OP from your story it sounds like she has both manipulated and deceived you (or you at least feel she has) and you have gone along with it all out of fear of not seeing your child. That is a sad situation. Ppl may bag you out but your feelings are perfectly valid. You are allowed to feel hurt and frustrated by this.
It's concerning that she refuses to discuss these issues with you at all - you need to find out why she went off the pill and why she refuses to work now. You have unresolved questions and they need to be answered. Perhaps she is overwhelmed looking after your child, perhaps work on top of that is really daunting, perhaps she's too proud for centrelink but you won't know until you force her to actually tell you. And the only way to do that is going to be via an ultimatum. Don't indicate the relationship will end - but perhaps what is going to happen to your assets if things continue as they are. What you might have to sell up or do without. How it's making you feel - that you want counseling either together or by yourself. Perhaps say you need some time to yourself - a week staying at your Dads house maybe, that you're that overwhelmed by all the work you're doing, it's too hard, you need a break. If she wants a break too, suggest you and your dad can look after the baby for a week since he is bottle fed a and she can also have some time to herself and a break.
If you find cooking and cleaning overwhelming, don't do it. Let it slide for a while. Get everyone to fend for themselves a few days.
Looking after a baby can be a TONNE of work. But I know how much stress my husband has felt from working/cleaning/toddler/me pregnant, and that's with me working 6 hrs a day 5 days a week. I don't think it's much to ask that she pushes a bit harder to lighten your load, ESPECIALLY if she won't give you a reason why she can't.
But sadly, if this doesn't work, and if you don't love her I can't see these things carrying you through 20+ years of your child's life. Splitting up may be inevitable. My only comfort to you in that situation is that if you put the effort in, you will still have a good relationship with your child. My parents are divorced, my Dad is FIFO, therefore I have only ever seen him once a month. We are closer than my Mum and I have ever been, and I lived with her every day. He got me a mobile when I was 17 and used to ring me on it and talk all the time. Children aren't dumb. He will love you.