Can I buy a plan for my father who has a visitor visa and lives with me? Can I cancel the plan after 3 month ?

Yes definitely. It's under your name and paid with your credit card :)

Fizz is prepaid service, you can cancel any time.

There is no identity check to get Fizz, especially if you use a prepaid credit card for the payments, so there's no problem with having the account in any name you want. The only thing standing in the way of complete anonymity is the requirement to send the sim card to your home address. But even then I don't think the right name needs to be involved.

Absolutely!
Add him as another number to your account.
I am not sure you can use him as a referral and get some reward in the second month.

@Ed, no referral to add sim in same account. You can only get credit by adding new account (new customer).

You can definitely do it!!

Yes fizz is perfect for this purpose