The second indefinite article is optional when it is noun A and noun B, also native speakers would normally put 'mum' before 'dad', it is just a set phrase, 'dad and mum' is not wrong, it just sounds a bit unnatural. So 'I have a mum and dad' would work well. In this case you are not using mum and dad as names, but telling someone what relatives you have, so capitals not needed. If you were telling your sibling that your parents were at your house, you might say 'I have Mum and Dad', here they are being used as names

The first option could be natural in British English. Americans will always say "mom" instead of "mum." They are only capitalized if you use a certain name to refer to a person directly. This applies to any name, not just the words for family members. So nicknames that aren't their actual name will also be capitalized. When you call your mother "Mom," you are using it as a nickname, basically. Dialogue: Son: "Hey Mom, I need help with this." Mom: "Okay Son, I will help you." "Hey Timmy, come here!" "Okay Slim Jim, I'll be there in a second." Description: I have a mom and a dad. I have a father, mother, and one sister.

You should only capitalize when you are saying the word instead of their actual name. So if I say, “Let me ask Dad a question,” it should be capitalized because I’m using Dad as his name. It's like saying, "Let me ask John a question.” But if someone were to say, “There is a dad looking for his missing kid,” I’m not calling that man dad, I’m saying he is a dad, or that he is a man with a kid.

So to answer your original question, it depends on context. If someone asked who you live with, you could say “I live with Mom and Dad,” and the implication is that you are referring to your own parents by Mom and Dad instead of their actual names. Or you could say, “I live with my mom and dad,” and that wouldn’t need to be capitalized, because you’re not calling them by name. That would be like saying “I live with my John and Jane,” which would sound wrong. The same rule applies to the other terms you mentioned, too. If you are using it instead of their name in the sentence, capitalize. If you are not, don’t capitalize. Hopefully that helps!