1) Where can I get a list of existing, working, reliable proxies?
the only one I know if, is mrhinkydink, though it'd be good to have other options too. Otherwise, some people use TOR..
You can use Elite Proxy Switcher to change, like if one has gone slow.
mrhinkydink has some SOCKS ones up which is an innovation.. But he could stop his service any time, he did stop for quite a while recently, but then restarted.
don't get expectations high re speed and reliability. it's just relative to other lists. There is http://hidemyass.com/proxy-list/ but I think you'll find http://www.mrhinkydink.com/proxies.htm is faster and more reliable but it's relative and isn't likely to be like your regular browsing experience, certainly any one free proxy ip won't be reliably fast even for basic pages.
2) How can I tell if a proxy is honest - that is, not stealing unencrypted passwords, etc?
You can't really. but if it's a company you pay and you have their details then there's some accountability perhaps, and they're less likely to steal, but still you can't know for sure. Certainly not any free online proxy list.
3) Do proxies exist whereby my computer can use an HTTPS connection to the proxy (reducing the worry from item #2) but passes appropriate HTTP to the website I want to connect to?
maybe but still might not trust it..
4) I really only want to use the proxy for one particular web site. By using the proxy in, say, Firefox, I assume all traffic to any site then goes through the proxy? That's fine if I can use two browsers. Say I use Konqueror for my proxy using browser - will this not affect Firefox which I can continue to use normally?
They won't affect each other obviously.
You could use FoxyProxy to assign a specific proxy for a specific site. Or you could use elite proxy switcher to switch universally.
Use http://www.whatismyip.org/ to make sure you are browsing with that proxy