How can I use my own payment method (PayPal) in the App Store instead of the family payment method?

You cannot. Family Members cannot use their own payment methods except for iTunes gift cards.

If you have enough iTunes gift card credit, purchases will be charged to that before they charge your father's credit card.

The credit card may still need to be verified through its security number during a purchase though, and If "ask to buy" is enabled, purchases will still need to be approved by your father, but it should not charge his card.

Hello FirestormFox,

Thanks for visiting Apple Support Communities,

I understand that you are wanting to add a different payment method to your Family Sharing Plan. The family organizer - who controls and shares the payment method for the family - will need to add a different payment method that you can use. The following guide details the steps to take:

Change, add, or remove Apple ID payment methods

Best regards.

FirestormFox wrote:

That’s incredibly lame. I guess it was only made for actual children in mind, not adults.

Correct.

If I bought something using family payment method and I leave family group, do I lose it?

No. The purchase is tied to the Apple ID that initiates it. In this case yours. If you leave Family Sharing, you keep your purchases even though they were paid by your father's account. Everybody else in the Family Group will lose access to it though.