The first one is a mess.
You can figure out the three cases of Y, but also I'm not sure how you want to write YOUR Geometric.
It can be written two different ways.
The second is best done by figuring out the CDF of Z.
There are two cases, we easily have the joint density of (X,Y).
We then can figure out P(X+Y<z) for z in (0,2) and then (2,4)
where the integration is done over the 2 by 2 square
Now be aware, in the second case, it is smarter to use the complement.
And note that X+Y=c is a line cutting through that square.