Logic equation for truth table with don't care condition
Well, there are many way to find out the logic equation from the truth table. One of way (probably the easiest to learn) is by using K-map (although it will become fairly complicated and hard to deal with if you have more variable):
http://www.allaboutcircuits.com/textbook/digital/chpt-8/dont-care-cells-karnaugh-map/

I think the link above give you a very good idea on how to deal with don't care term in K-map.

Other than that, if you have much complex equation, you can try doing quine mccluskey algorithm which you can find in the link below:

https://www.youtube.com/watch?v=B08vV3tIdag

For Problem 2 you mentioned,I am assuming Z is the output and X, Y are the input,to the best of my knowledge, I am not sure if you can do that because your second case X:0,Y:X,Z:1 is conflicting with your first case. It says that when x is 0, you don't care about the value of Y and the output has to be 1. But in your first case, you did specify the condition that when Y is 0 and X is 0, Z has to be 0. If that's possible to figure out the Boolean equation, please keep this posting and I will check it out.