I had one that was difficult to seal although the original had been good for 15 years. I found the "cultured marble" was rough in the seal area. I sanded it ; 50 or 80 grit , just to take out bumps, nothing fancy . Then the putty and rubber held fine. Apparently the original plumber used a lot of putty and torque.
Plumbers putty (or appropriate alternative) is to be used under the drain trim flange at the top of the assembly, to seal the drain at the sink bowl. NOT at the bottom.
The bottom of the drain assembly should seal to the bottom sink hole by use of a tapered rubber gasket, no putty.
It looks to me like there's plenty of available swing in your trap assembly to get better alignment. Swing the trap U around to the back and bring the other bend forward. Try various positions until you find one that works.
The leak is because the nut isn't tight enough. There's no magic to these things--just get a channel-lock pliers on it and give it ...
The main problem I see with this (otherwise fairly standard drain pump arrangement) is that you're now trying to run two things into the washing machine standpipe. That's not up to code, and swapping the hoses back and forth gets old very quickly. If you put the pump on the sink drain, and drain the washer to the sink, then you're back to one thing in the ...
For future reference, I was able to solve the issue of kinks in the snake by taking a pair of needle nosed pliers on either side of the kink and bending it back into place. This isn't perfect - there is still a bit of kink that remains, and it was still a bit time consuming to root through the pipe far enough to clear the obstruction, but it did work.
No, I don't consider that repairable. Enough metal corrosion has occurred that entire assembly is suspect and at risk of failure. The good news is that it's a fairly easy repair if you have access. If you'd like help with that, post some photos.
No, the metal parts won't damage your pipes. They're probably already out at the street.
I would agree the extra loop will create increased resistance and possibly not pump as well.
I would shorten the hose if nothing else to reduce the back pressure on the pump.
I know my washer has a check valve at the exit of the pump so it should not let dirty water flow back but I would think eliminating any flow restrictions and a longer hose is a ...