OP, yanbu. Sounds like you're having a bloody hard time of it. Poor you 
Being brutally honest, she's chosen to have sex, and to keep a baby despite still living at home and having no money or means of her own. So she should appreciate she's got to live in your house and with your rules.
I don't buy all this 'its only 2 weeks after the baby' ffs how do you think single mums manage? With DS1, admittedly I was 26 not 17, but I had no partner or family, and therefore had to get on and do stuff because no one was going to do it for me. That's how life is.
And it's not like the OP has asked her to cook a meal or clean the house, or do a weekly shop. She's been asked to empty a litter tray and put a bag in the bin, and chuck away some rubbish. It's 5 mins work, 10 tops. And as OP says, her mum, DGM, was there - I'm sure she could have occupied baby for a few mins whilst those tasks were accomplished. There's no excuse other than laziness/ oh well, mum will do it.
Make it clear she has certain jobs to do, they are non negotiable. That isn't being U in the slightest. I am sure once she sees it's either do that or move out and live independently, she will suddenly find the time and ability to pick up after herself.