As for the washing machine, the problem could be if the drain pipe is too small ( 1½" instead of 2"), if the drain is not vented properly, or just too sudsy detergent. WM drains are also susceptible to a buildup of lint. This will not be cleaned out much by running a typical 1/4" hand snake through the pipe. A commercial drain cleaning machine or jetter would be necessary. The enzyme treatments are a good idea. But they are a long term treatment; don't expect overnight results.
The T/P valve is just screwed in. You will need a good size pipe wrench ( 18") and not just a pair of pliers, to remove it. The new valve may already come with some white thread sealing tape on the threads. To this, you could add some paste pipe joint compound. There are two types of valves....short shank, usually used on the top of the tank, and long shank, usually used on the side of the tank. You could take out the old one and take it with you to the plumbing store to match it.
I gather that there is a pipe dropping straight down from the TP to a spot near the floor where you have the bucket. This pipe can be unscrewed from the T/P valve, and reinserted into the new one. If this discharge pipe has and bends and turns, then it may need to be cut, and then resoldered.
By the way, it does sound like just a leaky TP, BUT, if the water pressure in your house is too high ( due to an old, failed pressure regulator) this could cause the TP to drip. So, get an inexpensive pressure gauge and check the pressure at the washing machine tap before you replace the TP.