Yes, you can still use your printer if you cancel HP Instant Ink. The only difference is that you will have to buy the ink cartridges yourself.
If you cancel your HP Instant Ink subscription, the service will stop and you will not be able to print.
This is a tricky question. There are many ways to trick HP ink cartridges but I’m not sure if any of them are considered “tricking” them. If you don’t want to spend the money on HP ink, then you can buy a different brand that is cheaper or find a way to refill your old cartridges with cheaper ink.