I only know a little bit about this, so for your sake I hope someone more knowledgeable comes along and adds to this answeralong...
As for your first question:
It is not hard to see that the tensor product of any two central simple algebras over a field is again a central simple algebra over that field: see e.g. Theorem 79 of these notes. So for any field $K$ and any infinite cardinal number $\kappa$, the set of isomorphism classes of central simple $K$-algebras of cardinality at most $\kappa$ forms a commutative monoid. I don't know what can be said about the structure of these monoids.
However in this case it's not clear to me that there is some natural equivalence relation to impose here such that the quotient under this relation forms a group (as in the finite-dimensional case). One perspective on Brauer equivalence of finite dimensional CSAs is that it is a special case of Morita equivalence. I would tend to doubt that considering infinite dimensional central simple algebras under Morita equivalence will give a group: perhaps some expert can confirm this. Failing this, it seems that the burden is on you to come up with some natural equivalence relation and explain why it generalizes the Brauer group in the classical case. On the other hand I know people who like to think of categorical analogues of the Brauer group, so possibly this is a fruitful way to go here.
As to your second question:
No, definitely not. Among other things, Wedderburn's Theorem shows that any Artinian simple algebra is "almost" a division algebra (in fact is Morita equivalent to a division algebra). But the class of non-Artinian simple algebras over a field $K$ is far larger than the class of division algebras over a field $K$, and there is no reasonable classification of one in terms of the other. For some examples of this, see Chapter 1 in the notes linked to above, and then see Lam's First Course on non-commutative algebra for much more in this direction.
I have nothing to offer on your third question.
