I suspect the keyboard is your sole problem at this point... I have an early IIe which got a full cup of coffee dumped on the keyboard. After several cleaning attempt (including a run through the dishwasher) it still has problems with repeating keys as you describe. What happens is that the keys don't spring up properly after being pressed so the automatic repeat feature kicks in. Sticky keys can also cause problems at startup, like booting into self-test mode and so forth.

I'd look around eBay stores for a new platinum keyboard, they seem to be fairly common as NOS.

As far as monitors go, it really depends on what you plan to do with your IIe. If you're mainly into games then the Colour Composite is the better choice. The Monitor II is a monochrome (green phosphor) monitor, so it's not great for games, but much better for text...