You seem to have an optimism which, in the long run, might be misleading. Just because a person has been successfully in your life for 5 months-- that does not make a commitment. Even when you 'know'.
I'm guessing you also found the father of your child very attractive until whatever happened which changed your opinion. I mean, you did think enough of the guy to make a baby with him.
Personally, if you were my friend, I'd tell you to get your primary life (child, ex-boyfriend) straightened out. New relationships are already challenging enough. Imagine cohabitating with this guy, moving out of your stable living situation and then sharing a living and expenses, a financial situation with him. It's not as easy as it sounds, even without a kid involved.
I'd strongly suggest waiting to live together until you are both certain that you will be married in the immediate future. This will financially protect you. If this guy is for sure he wants to live with you, then marriage should be on the table as an option.If he isn't sure he wants to marry you, then I think it's okay to cool things off and take things a LOT slower. Or you will end up like a lot of people in your situation do-- in court, asking a judge to hash things out. Moving in together is a big step. Introducing your child to a new male figure is a big deal. I'd seriously ask yourself what the endgame is in all of this. How do you see your family's future? The last thing you need to do is make a baby with one guy, find that the relationship has deteriorated, and then, find another guy to play 'daddy' to your child. It's not fair for anyone. If he's committed, then let him show this over time. There's no rush.
And no, no Father's Day card. As an adult, I've sent my stepmom mother's day cards from time to time, but that was/is MY choice, not something an adult imposed on me.