Can a website still log my IP address etc. if I use a proxy?

Fredrik pretty much nailed the exact answer you're looking for - though I'd suggest using the an anonymous web network such as TOR.  I use it specifically for anything 'privacy critical'. https://www.torproject.org/ As per the website: Tor prevents anyone from learning your location or browsing habits. Tor is for web browsers, instant messaging clients, remote logins, and more. Tor is free and open source for Windows, Mac, Linux/Unix, and Android

I agree with many of these other answers, but I have to say that yes proxies and even TOR are helpful, but they do not provide absolute anonymity. My set up is: Hotspot VPN (premium, but the do have one of the best free versions) and TOR this way my computer connects to the VPN and the VPN server then connects to the TOR network making my IP just about impossible to trace, unless you were to seize it from the VPN company.

If you use an anonymous proxy, your IP address won't show.  The only IP will be that of the proxy. http://www.privateproxysoftware.com/Blog/identity-theft/why-use-a-proxy-server/

if you use transparent proxy, the website will log your real IP address, not the proxy IP. many of the web proxy lists contain transparent proxies. if you'll use a highly anonymous proxy, the websites you visit will log your new external IP address aka the proxy IP address. my recommendation is to check every time you use a proxy, your real IP address - http://whatismyipaddress.com/?u=TRUE also, for security reasons i use only paid private proxy servers... i care a lot about my privacy, my email addresses or my credit card info... check out http://ninjasproxy.com/private-proxy.