Are these terms as informal as they are in English?
I answered a simliar question just recently, you'll find what I wrote and some more here: https://www.duolingo.com/comment/5848211
Tack så mycket! That was very informative!
Is there a childish way to say mum and dad, like mummy and daddy?
I don't believe so. A child is most likely to use "mamma" and "pappa" just like an adult.
I wish there were more "pronounce these sentences and we'll tell you if they sound acceptable or not" sort questions in this Swedish course. They were my favourite!
What's the usage of this compared to "mor" and "far"? These are the words I have learned from other sources.
Zmrzlina's link above explains.
Probably it could be : mothet and father
I feel like a baby because I got this one for speaking.
In American English people use the terms mama and papa. I'm surprised they're not accepted as correct here.