Plumbing – How to extend this waste drain for new bathroom sink

Experience tells me that at least some models of washer won't hold water unless the drain hose extends above the water level of the wash basin. I can't quite figure out what you're trying to say about extending pipes, but it should be safe to have the water drain out of the washer drain hose at its maximum height.

Eventually, the drain hose can be high enough that the water column overwhelms the pump (or hoses), but you'd have to be doing something bordering on insane to reach that. Every foot of height in a water column adds .43 psi of water pressure, so even a hose rising 15 feet above the washing machine will provide less than 7 psi of back pressure. I can't imagine a washing machine pump which doesn't handle that.

You probably won't be able to replace just half of that trap as different manufacturers use different threaded connections at the union. Assuming the "lock ring" you referred to is the union nut (the large nut connecting the two halves of the trap), hand tight is not enough so tightening it carefully with a tool was appropriate. Unfortunately it is common for the nut to crack, causing a leak at the union.

I would cut the black horizontal ABS pipe downstream of the trap and replace the trap with an adjustable P-Trap. This will require a glue joint, but will leave you in a much better position for future repair or replacement when necessary. Use only ABS cement.

Remember, hand tight is not enough. Use adjustable pliers to tighten (but not over tighten) all slip-joint fittings.