Ashley I know you want the magical words to sway your mom, I wish I had them. This is your moms house, her rules, her life and while you have been gone, she has learned life without you around. She has a routine in which you about to disrupt. I speak as a mom and grandma and while I love my kids and grandkids to come visit, moving in would be another issue. It's true we like our ways as we age and really she is working two full time jobs, she is tired. Having that change in her house will stress her out, it's normal. While she loves you and welcomes you home, the dog is another chore to her.
I will speak as a mom in her position and try to give some ideas. Try: Mom, please give it one month and see how it goes. Make no promises, just make it your quest to make your moms life easier, no contracts. Those are useless with moms, she knows you and wants to trust you on being a adult. Instead give her your goals and plan on how to help her, keep the house clean while she is at work and show her you will pull your share, you sound very willing to do so. Show her you are actively looking for a job and do pay her rent, regardless of what you do at home. That will show her you are indeed a mature and responsible adult. Oh and don't use the word "Nothing" at all to her, she won't believe you and she still wants control of her house and her rountine.
Don't go our to hang with friends everynight, be home and responsible before you start dating and hanging with friends and only do that one or two nights a week. Be a example to your brothers, I don't know thier age but it really doesn't matter.
If you can, introduce the dogs on neutral ground, sick dogs might not accept a young dog to upset thier routine.
I had to move back home twice for a short time when I was younger and I didn't have animals, I had a young daughter. It was hard, I did upset thier routine and did everything I could and did pay them rent. I respected thier rules even though I had been on my own and married. Thier rules were a little stricter than how I lived my life, it was hard but I wanted to keep the peace.
We love our children, I wish you best of luck. I really wish I had a easy answer. I don't know you or your family, so I just spoke as a woman who is most likely older than your mom. Good luck, maybe someone else has more ideas, do let us know how it goes.