Several points (and I apologise for employing Heideggerese in these points; I hope they are clear enough):
Its important to realise that Dasein is in no way equivalent to consciousness. Dasein is defined, in Heidegger's formulation as "that being which in its being is always concerned with its being." Yet, in its everydayness, Dasein is unaware of this question or of this concern. Dasein thus names something prior to consciousness, the ground on which consciousness is founded.
(Addressing the aside): since Heidegger acknowledges that there are beings that do not have the character of Dasein, it seems that there are beings which cannot, in principle, pose the question of being. Given the famous formulations of The Fundamental Concepts of Metaphysics that a stone is worldless, an animal is poor in world, but man is world-forming, it seems likely that Heidegger intends that, at the very least, a stone is something for which the problem of being cannot arise. (Heidegger's thought is unapologetically anthropocentric).
Note that the "problem" of being is not something that arises for every Dasein. In fact, Dasein, in its everyday mode, tends to remain unaware of the problem posed by its being. This is why there can be a forgetting of the history of being. So even Dasein, which can potentially experience being as a problem for the most part does not.
The locus classicus for the distinction between consciousness and self-consciousness is not Heidegger, but rather Hegel's Phenomenology of Spirit, whose first two sections examine exactly the notion of consciousness and its relationship to self-consciousness. At the start of the self-consciousness section, Hegel has this to say:
In the preceding kinds of certainty [i.e., those of consciousness
rather than self-consciousness], the truth for consciousness is
something other than consciousness itself. However, the concept of
this truth vanishes in the experience of it. The way the object
immediately was in itself, as sense-certainty’s entity, perception’s
concrete thing, or the understanding’s force, proves not to be the way
it is in truth. Rather, this in-itself turns out to be a way in which
the object exists merely for an other. ... However, what has now
emerged is something which did not happen in these previous
relationships, namely, a certainty that is in parity with its truth.
This is so because in its own eyes certainty is itself its object, and
consciousness is in its own eyes itself the truth. To be sure, there
is also therein an otherness, but consciousness makes a distinction
which for it is at the same time no distinction at all. ... However,
consciousness is likewise that for which an other (the in-itself)
exists, and it is for consciousness that the object’s in-itself and
the object’s being for an other are the same. The I is the content of
the relation and the relating itself. The I is itself in its both
confronting an other and at the same time reaching out over and beyond
this other, which, for the I, is likewise merely itself. ... Thus, with self-consciousness we have now entered into the native realm of truth.
(Phenomenology of Spirit, §166-167)
In other words, there are, for Hegel, experiences of consciousness that are not the same as the experience of self-consciousness, and indeed, a kind of "certainty" attached to these. Self-consciousness names, for Hegel, the particular level of consciousness at which knowledge, rather than certainty is possible, i.e., on which the thing-in-itself is experienced in-itself. Note, however, the double-motion implied in self-consciousness, for self-consciousness is never merely a self-satisfied consciousness, but a consciousness of consciousness as such. Self-consciousness always points beyond itself towards that which it is conscious of, and so it maintains the general character of consciousness, i.e., that all consciousness is always consciousness of... .
This, in my view, gets at the important character of the distinction between consciousness and self-consciousness, at least as it gets carried on into the early part of the phenomenological tradition, i.e., that self-consciousness is something of a second-order form of consciousness, a consciousness of being-conscious-of-... . However, I think that such a description of self-consciousness leaves open the question of "self," i.e., that in self-consciousness what self-consciousness is aware of is not, perhaps, the self-in-itself, but rather the self-in-its-consciousness-of-..., leaving open the possibility of a fragmented, rather than unified self. Of course, such considerations are not possible in the most natural readings of Hegel, nor of the early phenomenological tradition, both of which consider all consciousness to fundamentally uncover a self-consciousness, and hence a self-in-itself.