What happens if someone is served anullment/divorce papers as well as custody papers for signing over the rights to their child and they don't get a lawyer, or respond by the date given in the papers?

Normally they will lose the cases by default, the court will give the other party almost anything they ask for that is within legal limits. They will have no idea of your income and will award child support based on what the other party says.

So you basically give them a blank check to get what they want without fighting for it, or having a say in it.