To show the 0-matrix is in U, you need to verify that:

A0B = B0A, can you do this?

For your next question, suppose that:

AXB = BXA and AYB = BYA.

You must show that A(X+Y)B = B(X+Y)A.

Start by expanding the LHS to get:

A(X+Y)B = (AX + AY)B = AXB + AYB. Do the same for the RHS. What can you say about: A(X+Y)B - B(X+Y)A ?