I wonder how existing major religions follow modern cosmology.
The question is excessively broad, even when restricted to major religions. E.g., Christianity is a "major religion", but within its different denominations, and even within the members of a particular denomination, there are different positions with respect to the conclusions reached by modern science, depending on the understanding of "biblical inerrancy":
Those who understand inerrancy as applying in exactly the same way to all subjects and in all Bible books, usually called biblical literalists, reject most of those conclusions, holding young earth creationism and sometimes even geocentrism.
Those who hold that the purpose of divine Revelation is to teach the truth which is relevant to our salvation, and not to teach natural science, profane history, or other forms of merely worldly knowledge for their own sakes, have no problem with those conclusions, as long as they are true scientific conclusions and not unfalsifiable hypotheses like the multiverse, eternal inflation, and the like.
I will answer the question from the viewpoint of a Christian in the second camp, who believes that the Creator spoke to humanity through a Revelation along human history and through the physical universe, and in both cases spoke the truth, so that if it seems (in a scientific sense, as in a plain sense it seems that the sun moves around the Earth) that events in the universe were a certain way, they were actually that way. If the narrative of events in the book where the divine Revelation was recorded differs in its details from what we can infer from the observation of nature, it is because of divine condescendence to the low degree of scientific development of the original recipients of the Revelation, and because, as said above, through that Revelation God did not intend to teach things in no way profitable to salvation.
In this framework, the actual case is not that Christianity "follows modern cosmology" but the other way around, as noted in Jayson Virissimo's answer and stated by Robert Jastrow as early as 1978:
At this moment it seems as though science will never be able to raise
the curtain on the mystery of creation. For the scientist who has
lived by his faith in the power of reason, the story ends like a bad
dream. He has scaled the mountains of ignorance; he is about to
conquer the highest peak; as he pulls himself over the final rock, he
is greeted by a band of theologians who have been sitting there for
centuries. [1]
Actually, the concordance of modern cosmology with the Genesis account of creation is much better now than it was in 1978. To be able to perceive that, one must be familiar with inflationary cosmology, as introduced e.g. in [2]. Noting that inflation does not leave any observable "imprints" or "signatures" of the time before the inflationary epoch (IF in fact there was such a time), it is perfectly consistent with current science to postulate that the universe began to exist at the start of the inflationary epoch, empty of matter and radiation and with only the "inflaton" scalar field that drives inflationary expansion. This hypothesis fulfills a basic requirement of science: that the initial state is maximally simple [3]. Let's now see the concordance between the biblical narrative, quoted verse by verse, and this hypothesis.
Day 1
1:1 In the beginning, God created the heavens and the earth. = Creation ex nihilo of both the spiritual (heavens) and physical (earth) universes. Alternatively, verse 1 could be a title for the narration in ch. 1.
1:2 The earth was without form and void, and darkness was over the face of the deep. = In the inflationary epoch there was neither matter (therefore the universe was void) nor electromagnetic radiation ("light", therefore the universe was dark), but only the "inflaton" scalar field. To note, a closed (finite) universe does not have an initial singularity, since the scale factor at t=0 is nonzero and finite, and starts at rest.
1:3 And God said, "Let there be light," and there was light. = At the end of the inflationary epoch, the inflaton scalar field decayed in the particles of the Standard Model (event conventionally but, in this hypothesis, improperly called "reheating", because there was no previous "hot" state), including the photons of the electromagnetic radiation, i.e. "light". These particles were initially mixed in a homogeneous hot plasma, in which photons were constantly colliding with protons and electrons, so that the universe was opaque, a "luminous fog".
1:4 And God separated the light from the darkness. = The ongoing expansion of the universe caused the gradual cooling of the hot plasma to the point in which protons could capture electrons to form electrically neutral hydrogen atoms (event conventionally but improperly called "recombination", because there was no previous state when protons and electrons were "combined"). Shortly after, photons ("light") decoupled from matter and started to travel freely (event called "photon decoupling", properly for once!). At that time, part of the initial electromagnetic radiation was still in the visible region of the spectrum, but the greatest part had alreadly redshifted into the infrared.
1:5 And there was evening and there was morning, one day. = As is well known, in the Israelite reckoning of time a day starts at sunset, so that each day is comprised of evening/night and then morning/daylight. This was strictly fulfilled in the "first day", when the universe started to exist in darkness ("there was evening"), and then "there was light" ("there was morning").
Day 2
1:6-8 And God said, "Let there be an expanse in the midst of the waters, and let it separate the waters from the waters." And God made the expanse and separated the waters that were under the expanse from the waters that were above the expanse. And it was so. And God called the expanse Heaven. = This reflects the initial formation of structure, during which dark matter started to collapse into the marginally denser regions by way of gravitational attraction, attracting in turn ordinary matter and giving origin to dark matter halos and then galaxies, leaving a void space ("expanse") between them.
1:8 And there was evening and there was morning, the second day. = Since the ongoing expansion of the universe causes a continual increase of the wavelength of the "background" electromagnetic radiation (that which started to exist at "reheating" in day 1), very shortly after "photon decoupling" that electromagnetic radiation went completely into the infrared region of the spectrum, so that the universe went back into "darkness" (from the viewpoint of hypothetical human observers) for at least 400 million years (My) ("there was evening"). Then, after the formation of structure, the massive and short-lived first-generation stars (Population III, or extremely metal-poor stars) were formed and started to emit visible light ("there was morning").
You cannot get better concordance than that.
Now, the main rival to inflationary cosmology is cyclical cosmology, which posits that the Big Bang was really a Big Bounce from a previous contracting universe. That hypothesis is inconsistent with the conclusively established finding that the expansion of the universe has been accelerating during the second half of its history [4] [5], so that anyone holding it must postulate:
either that some presently unknown mechanism will change the currently accelerating expansion into decelerating expansion, stop, and then contraction,
or that something changed between all previous instances of the universe and the current instance, whereby all previous instances decelerated their expansion and reversed into contraction whereas the current instance will expand for ever.
Either postulate can be described with just two words: fanciful thinking.
Answering now the rest of your questions:
"Is God a god of the planet Earth, or of the whole solar system, or of the entire universe?"
God brought and sustains into being everything which exists outside Him, i.e. the entire universe and any other universes He might have created. (To note, the multiverse hypothesis, even though it is not science because it is unfalsifiable, is not strictly incompatible with Christianity.)
"If he is the god of the whole universe, is he also the god of other hypothetical alien species?"
Yes, obviously.
"The common conception we have of god is anthropomorphic;"
Whoever those "we" may refer to, they certainly do not include Christians. The God that Christians believe in is the Subsistent or Absolute Being: one, absolutely simple, absolutely infinite in every perfection, spiritual, eternal and immutable. That's hardly an "anthropomorphic" conception.
"Christ (according to Christianity) is the son of god;"
Jesus Christ, according to Christianity, is the Son of God Who assumed a created, finite, human nature. Before and after assuming that human nature, the Son of God Is eternally the same and only Subsistent Being as God the Father Is. So the Incarnation has nothing to do with anthropomorphism.
"can god appear on Earth as a man but on another planet as an alien, where an alien species exists?"
Yes, the Son of God could also assume an instance of the nature of another rational species living on another planet. Which, for a Christian, is an argument for the unlikelihood of the existence of such species, on grounds of economy of Incarnations (and of Passions, if the members of that species have sinned).
If God is god of all the universe, then how has he chosen the earth in
the billions of galaxies and billions of planetary systems?
I interpret that the intended question, when accurately stated, is "how has He chosen ONLY the earth", because otherwise it does not make sense. Thus understood, the question is equivalent to "Why did God create so many billions of galaxies if He was going to create life in only one planet of one galaxy?" This can be answered on a strictly scientific basis, encompassing the fields of biology, astrophysics and relativistic cosmology.
From biology we know that life requires a certain concentration of heavy chemical elements. From astrophysics we know that present heavy chemical elements were produced by past supernovae and neutron star mergers. From the previous two points, the appearance of a planet with a chemical composition which is suitable for the development of life requires that the time elapsed from Big Bang be greater than some minimum value, since Big Bang Nucleosynthesis produced only hydrogen, helium and lithium.
From astrophysics we know that both ultra high energy (> 10^18 eV) cosmic rays and gamma ray bursts have extra-galactic origin. Therefore, a higher large scale matter density, since it means closer distances between galaxies, would imply a higher incidence on Earth of these highly dangerous particles and radiation. Thus, the good of living beings requires that, at the time of their appearance, large scale matter density should not be higher than some maximum value, which we can safely take to be the current value.
From the Friedmann equations of relativistic cosmology, assuming a closed (i.e. finite) universe with cosmological constant whose expansion starts to accelerate at t=ta, we derive the following inequality, where Rho_m(t) is the combined density of dark and baryonic matter and a(t) the scale factor:
Rho_m(ta) a(ta)^2 > c^2 / (4 pi G)
In general, at any time, if large scale matter density cannot be higher than a certain value, then the scale factor cannot be lower than a certain value.
If we add the requirement that the scale factor at time t_life when intelligent beings live is 1,78 the scale factor at time ta when the expansion started to accelerate, so as to enable those intelligent beings to conclusively discard cyclic cosmologies when they become scientifically and technologically advanced [4] [5], we eventually derive this inequality:
a(t_life) > 1.18 a(to) sqrt[-Wk(to)]
Where a(to) and Wk(to) are the current actual scale factor and curvature density parameter.
Thus, for Wk(to) = -0.001 (Planck 2013):
a(t_life) > a(to) / 27
And for Wk(to) = -0.0001, the smaller magnitude which can in principle be observationally distinguished from zero (Planck 2015):
a(t_life) > a(to) / 85
So, you cannot even in principle say if the universe is actually more than 85 times larger than the minimum size it must have so that:
the incidence of cosmic and gamma rays on Earth does not exceed its actual values, and
humans are able to observe clearly the acceleration of the expansion of the universe in order to conclusively discard cyclic cosmologies.
References
[1] Robert Jastrow, God and the Astronomers (1978), p. 116; (p. 107 in 1992 edition).
[2] Ethan Siegel, 21 09 2017, The Big Bang Wasn't The Beginning, After All.
https://www.forbes.com/sites/startswithabang/2017/09/21/the-big-bang-wasnt-the-beginning-after-all/
[3] Sabine Hossenfelder, 22 11 2017, How do you prove that Earth is older than 10,000 years?.
http://backreaction.blogspot.com/2017/11/how-do-you-prove-that-earth-is-older.html
[4] David Rubin and Brian Hayden, Is the expansion of the universe accelerating? All signs point to yes, The Astrophysical Journal Letters, Volume 833, Number 2 (2016).
https://arxiv.org/abs/1610.08972
[5] Balakrishna S. Haridasu, Vladimir V. Luković, Rocco D’Agostino and Nicola Vittorio, Strong evidence for an accelerating universe, Astronomy & Astrophysics, Volume 600, L1 (2017).
https://arxiv.org/abs/1702.08244
