If I buy a game from PSN, will I be able to play it without Wi-Fi?
As long as it's downloaded and installed yes. Online multiplayer games require PSN but single-player games can be played offline.
Also Once you've downloaded them onto two different systems they get greyed-out on your download list until you either call Sony and ask politely or just buy them again. You can play them for as long as you want until some erases your user login data from the machine, then you just have to make a new user with the same login and they work again.
Edit: Since November 18th, PlayStation restrict sharing from 5 devices down to 2.
In short. Yes, with one exception. If you purchase a "free" PSN+ game, that game has a license expiration. You can play that game offline until the date of the expiration, at which point the system will want to connect to Sony to validate if the license has been extended or not.
If you look at the information on a game you will see the date purchased and if it has an expiration, that date is included as well. If the game has no expiration it can be played offline until your system stops working or you delete/deactivate the PSN user account from the system.
Yes, it is 'yours' for as long as PSN exists. With 'yours' I mean it is tied to the PSN account you used to purchase it.
You will be able to play it just as much as if you bought it somewhere else.
Once you've downloaded the game you're able to play it offline except the game requires an internet connection.