Sure you can. osCommerce comes as a .zip file that cPanel > File Manager can unzip. Fantastico is not mandatory -- it just makes life a bit easier if you can tolerate a fairly "vanilla" installation of a product.

I don't see why you couldn't FTP the files up to your server.

Since this is a shared MySQL server (assuming a Basic or Business plan), your databases and your userIDs will be prefixed with your account name and an underscore. Say your account name is spufus8 (usually derived from your domain name); your databases and userIDs would start with spufus8_. This keeps everyone out of everyone else's hair, but can be tricky if you forget this when doing MySQL coding in PHP or whatever. I haven't heard of LP enforcing any restrictions on names (so long as they're legal MySQL/Linux names) -- dunno what they'd do with an obscenity.

FTP will give you access to everything in your account (/home/ACCOUNTNAME/ and below). "World writable" permissions (such as 777 or 666) will cause problems (usually "500" errors) due to the security software. There's no real need to go beyond 755 or 644, but some software (SMF and osCommerce come to mind) seems to insist on 777 or 666.

If you can list any specific database name restrictions and FTP permission restrictions your previous hosts have laid on you, maybe someone here can tell you if they would apply here?