Im from sweden and i have never in my life heard fäder
But it is the plural of fader/far. But yes, pappor is much more common.
Is it a set rule that the plural forms of family members (specific (brother, mother, etc..)) to have umlauts?
Nope. Sisters don't: syster -> systrar.
When do i use mamma/pappa instead of mor/far
Mamma/pappa = mum/dad
Mor/far = mother/father