We do exactly what you describe.
We use Fedora 11, with a self-downloaded MySQL 5 instance (i . e. we downloaded a binary MySQL 5 install, and installed that instead of using FC11 packages) and a self-downloaded PHP5 instance (which we compiled locally).
To match our ISP (http://www.hetzner.de/
) I simply popped a phpinfo() into an online server we have with them and examined what it spat out. I then downloaded and compiled all necessary packages and setup parameters on our local Linux FC11 dev server to match the live server.
We now do all development locally on the dev server and upload the entire site once done. It works fine. So your idea is definitely possible. The only problems we have had so far were some minor library version incompatibilities as regards GD (when implemented in PHP's as a graphics library) - for the rest it was quite painless.
Most what you need to do is quite easy and MySQL and PHP have extensive forums and other users available on their BBs that will be able to help you to the nth detail. Alternatively, post here again with specific questions and odds are you will get quick answers.