Seems like everything is an excuse to her..
She can't work because she's bi-polar. I'm sorry, but she can still work at McDonalds :) She can still hold some type of job. Maybe not Corp. Executive, but she can still hold a job. Plus.. In this day and age, there are a lot of thing out there to help people with Bi-Polar Depression. I don't know if she is, or has looked into any type of medication, but it would definitely help.. Maybe even some support groups. Mental diseases I know are very important, but no longer are they a crutch.
She won't let you move in over a dog? I love my dog too... but I don't think I would not allow my boyfriend to live with me if I couldn't have my dog. She needs to tell her ex good bye. She needs stand up for her own relationship with you and do what's right. Allowing him to blackmail her with a dog is a bit ridiculous, and a lame excuse to not allow you to move in.
As for the kids.. I can understand that.. But who's the adult? Better yet.. Who's the parent? Her kids are going to dictate her relationship with you? I don't have kids, so maybe I'm not sure what the proper protocols are, but if she loved you.. she'd MAKE it work.
She's using you. In every way possible, she's using you. She's taking you for granted and treating you like you're an idiot. You're paying for a house you do not own. You should thank your lucky stars that YOU'RE NOT on the lease.. She won't allow you to live with her because of a dog. She expects you to trust her with her Ex who comes over as he pleases. This relaionship is turning into a complete disaster.
Maybe not break up with her.. but back off... back off completely. She needs to pay her own rent. She needs to buy her own food.. She needs to ditch the ex.. she needs to either take the dog, or leave the dog.. not letting this lag out. Seriously, how long does it take to save you for a deposit? Deposts are what... $800 at the most? I've never seen any animal deposit more than $800... And couldn't you take the dog until she can? She needs to find a job and keep it.. Even if it is McDonald's.. Because it shows she is responsible.. Even if its part time.
What type of advise would you give if someone was in this type of situation? Would you honestly tell someone else to just hold on and it'll be OK eventually? I bet not.