If I wash my tools (knives and mud pan) in my laundry tub/sink after use, will the drywall mud clog my drain pipes?
I think it would depend on the amount of compound you're putting down the drain, what size drain, how much water is used to flush the tools, and probably a few other factors I haven't thought of. Though typically it's not a problem, as long as you're not pouring a whole tray of mud down the drain.
Scoop out any left over mud from your tray/hawk, and throw it away. Then rinse all the tools thoroughly.
NOTE: If you're using premixed compound, never put unused mud back in the bucket. Once the mud is out of the bucket, either use it, or toss it.
Drywall mud's redeeming quality vs its better qualified plaster cousins is its solubility. Adding any water to drywall mud will slowly dissolve it. Add a lot of water or hot water to mud (in any state) will start to dissolve the material, quite quickly if water overpowers.
The only possibly issue is if you poured down a chunk that blocked things then put more stuff down the drain - in effect blocking your blocked mud. Otherwise water will dissolve whatever is stuck and it will be a non-issue. Should you put vast amounts down a drain? Only if you are confident that this blockage can't become blocked. If you have clean pvc you could probably dump bags of drywall mud powder down without an issue - I wouldn't recommend testing this though.
What is funny is that probably the worst possible thing you can put down your drain is cooked, soft boxed pasta. Once it hardens it sticks like mad and takes quite a bit of time before it softens.
In general I wouldn't go pouring gallons down your drain, however if you put some down just run hot water for awhile afterwards. Supposedly if you have already done it and even supposedly if a little but hardened, the hot water should still loosen it up enough to separate and flush down. I always wash my tools in my sink that have paint or putty left over on them and flush like I said afterwards and have never had any problems.