I'm not a professional plumber, but fairly experienced with Plumbing. From what I understand, a p-trap allows for a small pool of water that acts as a barrier between the nasty stuff down your drain, ie. poisonous gases and smells. It also can catch bigger stuff that shouldn't be going down the drain and block it in a generally easily accessible spot, versus clogging up further down the line. It is certainly required by plumbing codes and should the sewer/septic gases build up they can cause illness/death or cause an explosion.
Get use to smell of sewer gases, or install p-trap under kitchen sink.
You usually have to unmount it and get at the bottom.
The shape of the drain. A P trap drains out the wall. It looks like a P laid on it's side. An S trap drains in the floor. Again if you laid the S on it's side.
Probably rust and minerals that came loose when you turned the water off and plugged the screen in the spout. Unscrew the screen the the end of the spout and see if that doesn't fix it.