There are a lot of contentious issues with your scenario and I think everyone has been more or less very gentle with you. We've known you for awhile with your threads and the theme is generally one of an individual who is willing to please, open to new relationships and uncertain about her future overall. Most of those relationships unfortunately have been inappropriate. Your future still seems uncertain and there's always an element of danger or walking the line with you and I think you're deliberately either ignoring your reality or you're not seeing things for what they really are.
Your pup will always be a difficulty for you for as long as you're renting. I'm sorry to say that. I've had dogs before and they're not easy to have around while juggling other things, let alone renting an apartment. I'm not sure if I could do that. I'd suggest you find a second job to save and keep afloat but this renders the living situation for your dog inhumane or difficult also. I hope it's not left alone at home for long periods even now. An alternative is doggy daycare but that negates any increases in income you may be making with a second job. I still don't know why you accepted a pay cut but perhaps my question was missed by accident on page 1.
I still don't find it acceptable that someone is willing to date you exclusively and seriously and yet charge you for a room. It tells me that he's not in the right frame of mind and neither are you. You're both looking for a quick fix...it feels like desperate times. And again, there's the dog to worry about on top of your own safety and stability. If this situation deteriorates I have a feeling it will deteriorate quickly with or without a contract. I also have to ask how many rooms he's renting out besides the one he's offering to you. How safe are you in that house? If he's looking for a roommate, his language shouldn't imply that he's "willing to open up a room". It sounds like he's doing you a favour but what he's doing is self-serving too. He feels slithery like an eel.
I think your mum might be worried for you. I would be worried for you if you were my daughter.
I cannot tell you what to do and it is indeed your future ahead of you. I'd encourage you to think a bit more critically of the options you've laid out before you and consider making some sacrifices that may be difficult but not impossible. You may want to consider rehoming your dog or looking into doggy daycare options and a second job that's feasible. You're trying to have your cake and eat it too and also keep tabs of what someone else is bringing to your table. It's not working. Try and look at one thing at a time starting with the most urgent and prioritize. This may or may not make sense to you and some parts may hurt you. It's not intentional. I hope you find a safe place to live and continue to try and make peace with your family. I feel like you have not healed from the difficulty in your past with your dad leaving. I'd seek therapy and a safe place to unpack that.