How are moral decisions made in such contexts? For example, if x believes old people are more important and y believes young people are, how would they make a decision where they would have to choose one life over another? This is just one example, but in general, is it best to choose randomly in such circumstances? I mean, it's impossible for one to say to the other you are wrong; because it is not objective or logically deduced which life matters more. How then?
know understand right from wrong the same way we understand anything else -- by coming up with a mental model for it. And in general, that model is as simple as "act in the best interests of others".
It is just as obvious, however, that no one can be comfortable using it until they achieve a sufficiently deep understanding of themselves, their lives, and the world they live in.1
Or, as Socrates would put it, "Knowledge is the only true virtue".2
1 starting with a profound appreciation of the pavement job they've been doing on that one-way road
2 in case you're wondering if it's even realistic to expect that a person can know enough, you can stop right there
There are multiple assumptions in your question which, when examined, may not be valid. Examine them, and the nature of the problem you are wrestling with changes. Note that identifying, then questioning assumptions, is an essential core skill for doing philosophy.
The first questionable assumption I saw was that you seem to think that the world, and pretty much any course of action in it, can be established "objectively", and that "subjective" is an unacceptable/bad appellation. But everything we know about the world is knowledge we have gained subjectively, through our individual personal senses, and then processed and evaluated through our subjective minds, and subjective reasoning process. We don't actually have access TO anything "objective". We INFER an objective reality, based on inter-subjective consensus -- but that isn't actually "objective". Instead of working with "objective" we have instead developed a pragmatic set of tools and methods to live with everything actually being subjective, and only cross-checked to some degree with other subjective evaluators.
What the subjective/objective discussion shows, is that there are pragmatic, practical skills and techniques, that lead to better success in this world and this life, and plausibly to better understanding of an inferred objective reality. Wisdom and skills matter then -- and are what one should be looking for in life, rather than "objective truths".
Applying our subjective circumstances to moral questions is one of the most difficult, and most important, of skills to develop.
There are multiple authors, over millennia now, who have written a lot of useful and interesting thought on how to do morality wisely. The best of the ancient writing is often included in "wisdom traditions", such as are in Aesop's fables, or the precepts of Confucianism. These treat morality not as a theory, or an abstract reality, but as a set of skills and precepts that lead to a wise/good life.
There are many other approaches to morality. Sociological ones, look at social stability and community consensus. Biological/evolutionary ones look at what promotes human species survival, or sometimes Gaia survival. Moral Theory approaches start from consciousness, derive welfare from that, and assume that an objective moral reality compels our adhering to it in our treating others who have a welfare. This approach has many variants from there, including Utilitarian, Rights, and Virtue ethics.
That these many approaches don't agree, is typical of the information and theory uncertainty we have to deal with and live with in an uncertain world. What is our MORAL obligation with moral questions? That is UNCERTAIN! But the recognizably WISEST among us have sought to come up with better and better sets of moral systems and advice. As a philosopher, one should seek out that advice, sort it by your own best judgement, and then try to implement the best of it in one's own life.
(1) It is actually the case that everyone has different moral values?
Moral values diverge according to groups, societies, social classes, religions, political parties, communities rather that according to individuals.
(2) The fact that people think differently does not imply that everyone thinks correctly. Moral values are the values people should accept (even though, in fact, they do not). In some sense, what people think has nothing to do with the truth of the matter. (Though, when it comes to morals, things are more complicated. Due to the fact that an ethical code is aimed at being applied, and therefore, should be accepted by most people in order to actually "work".)
(3) Even though it may not be possible to reach a consensus regarding the Good (the good life, the good life plan) it may be possible, and even necessary, to reach a consensus regarding the Right (what is just). It is reasonable to say that everybody should admit that:
a society is just iff it allows everyone to live according to his/her favorite (reasonable) conception of the Good
So, the very fact that we diverge as to the Good forces us to reach a consensus as to the Right.
Note : Rawls, A Theory Of Justice and its applications to health care problems , and more particularly, to the problem of "triage". https://pdfs.semanticscholar.org/11c3/bf025673a033add9a148cf1ef446bf077ebb.pdf
(4) As a matter of fact, wrong and right are decided by law that regards vital human interests. Laws leave room for ethics regarding less important questions.
Humans will generally tend to take a position on any subject based on factors such as familiarity, upbringing, and what is better suited to themselves. There is no right or wrong answer necessarily. Just an answer that is right by the standards of how / what lens and individual views the matter through based on the knowledge and experiences they are subjected to. I would suggest that in the proposed "Young v.s. Old" they compliment eachother and you cannot have one without the other. So it sounds more to me like the age old question of "What came first? The chicken or the egg?"