Concerning your question about MySQL. I'd have to check, but granting and revoking is done in the grant tables, under the mysql database. As far as I know, the root account is the only one with access to these. Therefore, the only was someone could grant or revoke access to your account is if they are authorized to do so (such as the sysadmin), or if they somehow found a way to get the root password. As long as the sysadmins know what they're doing, you should have no problem. The directories where table data is stored shouldn't be accessible from a normal account. Finally, the directory where the web files are stored should be owned by your user, and not accessible by anyone else (except root). Once again, these are both matters of the sysadmin correctly configuring the server.
If you want a seperate process, then you proably couldn't get one. At least not on a machine shared by multiple users. The only way to get your own would be a dedicated server, or co-location hosting. Both are somewhat expensive, though prices vary. But doing so gives you complete control over the server (root access to it, usually). You configure everything just how you want it. Unless you want to run an extremly popular site, or your picky about how your server is set up, you'd proably prefer a normal webhosting account.
