Yes, is is possible with wordpress. You can create custom pages with the funcitonality you want, through an extension.
I suppose you could yes, you would need to play a little with the configuration. When I say a little I mean you would have to literally rewrite allot of how wordpress work.
Also I have had my shared hosting accout hacked via a loop hole in WordPress, so I wouldn't advise you to use WordPress at all.
Josh, I wouldn't write wordpress off because it got hacked in shared hosting.
Shared hosting is usually more of the problem then the software installed on it.
Almost every account on shared hosting can be hacked, even if you use strict security measures.
ie: PHP usually runs as the same user on a single machine on shared hosting. So it can write to any file that it created, even on a different accounts.
If PHP can write to it, then anyone on the machine can write to it. It is the same for any running process on a shared host.
So when you're on a shared host, no matter how well you secure your site, if a single other site on that machine is vulnerable, your site is.