I heard rumours that after walking into a bathroom, nowadays, you're required to wash you're hands three times (as in, I've seen some Rabbi do this: after just walking into a bathroom to get something, he washed his hands 3 alternating times).
I also heard (and experienced) that if there's a partition between you and the actual bathroom, you can wash your hands there, even for bread and davening. I'm not sure though as to the extent of the partition required. In the bathroom at the shul where I go to, they have the netilas-yadayim cup in a sink that's inside the actual bathroom, with only a partition separating between it and the urinal, and the other toilet is right next to it (but that is fully partitioned off). Would something in that case still be considered a valid partition, even though it only blocks the urinal from one side, but you can still see it in the front, or does the partition have to block it from all directions, or is only a closed door sufficient?
Basically: what is the extent of the partition needed to wash your hands in a bathroom for bread or davening without any halachic concerns whatsoever?