Faucet handle removal
This is a NUT: 

Channel locks or a slip lock wrench will work. Remove the metal piece on the threaded pipe first.

For the top, I agreee with @jstola but it looks kinda like someone may have filled the bell housing with caulk (the white goo out the top doesn't belog there).
Once in a while, these housings have a small recessed hole with an allen nut inside; yours does not look like one of these, but if you see a small hole in it that looks useless, it probably is.
Also, sometimes these fixtures are also threaded on (and I believe sometimes they are a reverse thread), yours could very well be one of these. If it is caulked inside, you might need to (sensibly) apply some heat to get it to break/melt free.  Definitiely try the putty knife idea first, clean off all excess plumbers putty or caulk, and remove anything plastic. Dont cook the countertop.