Are deep rims more durable for a heavy rider? Since I weigh quite a lot (~115kg), and ride on bad roads with many holes and cracks, I am looking for rims that would be able to cope with bad conditions without getting out of true or breaking. My tires are 37x622, maybe I will go to one size wider in the winter.

My first instinct is that deeper rims should be more durable. Am I right? Are there some down sides to deep rims? Are they equally easy to true and put tires on?

For instance, which one would be better choice in my situation, DXT 337 or DXT 221
I'd actually lean toward a shallower rim, but the important bits are going with a high spoke count and either 3x or 4x pattern (both drive and non-drive sides). With a deeper rim, you are changing the angle in the cross patterns and reducing the length. This means that the spoke has to deflect more to handle the cross.

This is more noticeable with really deep rims, with the slight differences in depth for the rims you listed, the difference would not be that much.

I've built some race wheels and training wheels for a customer that's on the heavier side and competes in Ironmans. I've built the fronts with 3x both sides and the rears with either 3x both or 4x drive with 3x non-drive and haven't had any problems.

With your size and the road conditions, you will still need/want to true your wheels on a fairly regular basis, they will go out of true. If you get in the habit of giving them a quick check and true on the bike, you'll soon get where you are comfortable maintaining them.

Happy Riding!