1. If the pump is the same as mine (Flojet R3526 triple membrane pump, very similar in design to the Shurflo Aqua King): There's a valve inside the pump which prevents backflow through the pump. If that valve is tired (likely, if it's the original pump), it won't hold pressure after shutting off and water flows back through the pump. The pump is easy to take apart, so have a look. Might just be some pubic hair stuck in it. If you do replace this, consider changing the membrane as well while you're at it and give the whole thing a good clean. Parts can be found on eBay, although the entire pump isn't too pricey either.
The bodgy solution is to simply add a check (non-return) valve in the hose near the tray.
3. The chrome pole only has two ends, so it must come from either of those. The top goes into a ceiling panel, which is removable, so you would be able to see if it ran in there (unlikely). It's more likely that water simply seeps in at the bottom when showering and runs out later. This should be easily fixable by removing the pole and sealing it at the bottom (white butyl tape). Dry it out well and run some oil through to slow the rusting before you put it back in.
