I'll give your questions a try, Mark.
The "bleeder valve" you speak of is probably the manual lowering valve - it should be closed, or the top will not raise.
The plastic cap under the cabinet in the kitchen is indeed for the kitchen faucet. If you study things, you'll see the faucet probably sits higher than the underside of the cabinets when the top is lowered. Make sure your faucet is rotated so it will be inside that cup when the top comes down.
It sounds like you could be missing a section of what is fondly called the "stink pipe". I've not actually studied mine, since I'm having no trouble with it, but I suspect you need a section of pipe that will fit into the pipe at the ceiling or into the black tank at the bottom. When set up, I THINK the two sections of pipe slip into each other so that nothing has to be removed when the top is put down. I'd think the pipe at the top would be the larger diameter one. You may have had one or both "sawed" off. If so, they can be replaced with suitably sized ABS piping. You can probably fit new sections on the stubs that are probably still there using appropriate couplings.
I'd consider trying to fill the screw holes with epoxy (use the strongest stuff you can find), and then, after it's cured, pilot drill those holes and try reinserting the screws.
- Jack