Temperature:
Modern 3D cards get hot when working hard on a complex 3D game, but consume little power otherwise because of various power management techniques. They also have overheat protection features so a total meltdown isn't going to happen.
You can still find fanless cards. Newegg has a way to specifically find them in their advanced search. These cards typically have large heatpipe-equipped heatsinks and of course need at least some case airflow.
Fans on modern cards tend to be reliable in my experience. I have not seen a motor die in a long time. I have had cards with fan motors that get noisy however.
Compatibility:
When both NV and AMD went into the DX10+ era, they dumped a lot of backward compatibility features. Both companies' cards will have various problems with games using DirectX 6 and older.
NVIDIA cards tend to still have the best game developer support but AMD isn't at an obvious disadvantage anymore. I run both NV and AMD cards. There have however been occasions where AMD has completely dropped the ball. For example, RAGE was almost unplayable on AMD cards for 2 months because of OpenGL problems that were not resolved before game release.
Also, if you want to use any apps that use CUDA, obviously you want NV. There's also the NV Physx consideration, but frankly none of that stuff has ever impressed me so I don't worry about it.