Why are quantum groups so called?

Typically in math "quantum X" means a deformation of "X" which is in some sense "less commutative." So quantum groups should be deformations of groups which are "less commutative." Interpreting this is slightly tricky since groups are already non-commutative, but nonetheless they do have some "commutativity" built in which you can see either by noting:

1. The ring of functions on the group is a commutative ring.
2. The tensor product of representations of the group has a symmetric isomorphism $V \otimes W \rightarrow W \otimes V$.

You can use either of these to motivate versions of quantum groups. A quantum group is:

1. A Hopf algebra which deforms the Hopf algebra of functions on a group, but where the ring structure is non-commutative.
2. Something whose category of representations deforms the category of representations of a group, but where the tensor product structure is not symmetric.

For quantum matrices and related objects specifically, I heartily recommend the opening chapters of "Lectures on Algebraic Quantum Groups" by Brown and Goodearl. I hesitate to write any more details, since they do such a good job of it, IMHO.

Yes it can be understood as a quantization procedure. You are probably familiar with the first step to giving a quantization is to give a Poisson algebra. In this case you want the algebra in question to be better than the coordinate ring of the group, you want it to be the coordinate ring of a Poisson-Lie group.

Let's focus on the identity. Now I'm giving you a Lie bialgebra. Many examples come from looking at QR or LU factorizations. Here you see Etingof-Kazhdan.

I recommend Semenov-Tian-Shansky's lecture notes http://arxiv.org/pdf/q-alg/9703023.pdf

Edit: the deformation parameter has a different interpretation not necessarily $\hbar$, so be careful when you have multiple deformations.