GZip is a compressed version of the file, just like WinZip, Rar, etc. the .sql is the raw, uncompressed file.
To backup/restore using the way I'm talking, you'd need to have shell access to your server (SSH). It's a secure terminal that gives you command-line access to the box itself. If you don't have a direct SSH connection, a lot of hosts (using things like Virtuozzo) have a java-based SSH window that you can use as well.
If your host offers you neither, the best bet for you would be to backup your database, upload it, and then just have their support team run the import locally on the server. If they give you any grief about it, tell them that you either:
a) Need shell access to do it yourself.
b) Need another host that will allow you to SSH, or help you out.
PHPMyAdmin timeouts are super common, especially with big filesizes. The only other option you have would be to manually chop the .sql file into smaller bits and import them one at a time. And that, let me tell you first-hand, absolutely, positively SUCKS.
