If is is for a large 4" drain for example, the hopeful answer is "maybe" - that size may be strong enough to withstand disassemble/reassemble. But the best attitude to take when dismantling old metal Plumbing is: "It's probably gonna wreck it, and I'm going to have to replace it with a new one", because 95% of the time, that's exactly what is going to happen. Plastic plumbing is a lot more forgiving.
You usually have to unmount it and get at the bottom.
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.
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.