When I had birds that's precisely what they did. Birds typically like to be in the highest comfortable place in a room for instinctual reasons. In my apartments at the time, that meant the tops of virtually any open doors. (They also liked company so they'd generally follow me from room to room and sit on top of doors along the way.) You really don't want ceiling fans...

So, yeah. I made sure there was generally something on the floor below them. But the sides of the doors would need to be cleaned fairly regularly. I had cockatiels, and you couldn't "potty train" them.

Birds also like to chew things sometimes, so you needed to keep a close eye on them or realize your damage deposit was going to take a hit later.

The bird we have is a super social species so his favorite spot is on your shoulder or inside your shirt. He flies away to poop. We were able to potty train him so he doesn’t poop on us and generally avoids couches, beds, and such.

He does poop everywhere else though. Usually in a few common spots. Fortunately it doesn’t smell and we just wipe it with a paper towel.

Training him to poop in sinks had limited success. We aren’t patient enough.

As for windows, we keep screens on them. He probably wouldn’t fly away on purpose but when a bird gets spooked you never know.