What happens if I buy a paid game while it's free? There's this game on the App Store called Flying Hamster HD, before it used to cost like $6 or something now it's free.
If I delete the game I “purchased” when it's free and decide to re-download it am I charged if the price goes up?
When it's available for free and you buy it you get it for free.
It's impossible to charge you afterwards because when you decide to download it because it's free. You don’t get refunds if the price goes down or charged if it goes up.
Many apps become free over time, and that's one of the ways how I save a lot of money.
So in short, free = free (watch out for in-app purchases though, they are never free!)
To give you a better detail about updating apps:

It's free if they update the same app, not if they decide to put a complete new application in the AppStore.