Been thinking of creating a js/PHP/MySQL game, with multiple players / games happening concurrently. Is there any chance SQLite might be faster for something like this than MySQL? Keep in mind that I'll propably test this on a shared hosting server...
You probably don't care. I'd suggest that maybe it's not necessary to use a database at all, and a commercial gaming company almost certainly uses a custom server process (not a web server at all). However, you can do whatever you like.
In particular, you're very unlikely to have either enough data or players that you will actually notice. Gaming data are very, very small unless you start storing huge amounts of historical reporting data for previous games. And if you do decide to start doing that, you can host historical data on independent servers so they can't interfere with new games.
Writing every single action to the database, like you are suggesting you are doing, is usually a bad idea.
You would do much better writing a real server, and update the database only once a match ends.
My advice is to look at a PHP interface that is database agnostic, so you can switch it later.