How to Latex-print matrices as a bold uppercase letter?

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Ragarding the first question, you should understand that when you write:

sage: A = matrix(2,2,var('a', 'b', 'c', 'd'))

A is a Python name, that is a kind of pointer to the matrix object, but the matrix object doesn't know anything about which names point to it. For example, you can then do:

sage: B = A

Here B is just another Python name pointing to the very same object in memory as you can check with:

sage: B is A
True

So, if you want to type A.as_symbol(), there is nowhere in the matrix object that registered the fact that the matrix has "A" as a string representation.

We can define a function that finds one of the Python names of the matrix object within the globals() dictionary and pick the first one that appears (and that is not ugly like _42), but it is kind of artificial, like:

sage: def sym(a):
....:     for i in globals():
....:         if not i.startswith('_') and globals()[i] is a:
....:             return SR.var(i, latex_name="\Bold{{{}}}".format(i))

You have:

sage: latex(sym(A))
{\Bold{A}}

But of course:

sage: latex(sym(B))
{\Bold{A}}

Regarding the second question, it is a bit different, since the symbolic variable knows a string reprentation of itself, and you can even tune a latex one as follows:

sage: A = SR.var('A', latex_name="\Bold{A}")
sage: B = SR.var('B', latex_name="\Bold{B}")
sage: c = SR.var('c')
sage: A*B
A*B
sage: latex(A*B)
{\Bold{A}} {\Bold{B}}

sage: e = (A+B)*c
sage: e
(A + B)*c
sage: e.expand()
A*c + B*c
sage: latex(e.expand())
{\Bold{A}} c + {\Bold{B}} c