Generally, you will be on the losing end if you break a lease. If you have lots of documentation, your best bet is to tell them that they are not living up to the lease and say you wish to break it without prejudice (then get it in writing, clear as day). If not, then you can get your Attorney General's office involved - quite often they deal in landlord/tenant disputes. From there, you can try to get the lease broken. It will not be quick.
Breaking a lease is often more than just a single month of rent lost. Check the lease carefully, because quite often you are liable for the full lease. At the very least, up to the point that your landlord rerents the place and even then they will usually tack on charges for stuff like maintenance.
It is possible they know about your cat. All depends on the landlord if they will make an issue out of it. The cat would also be a reason you will not see rats or mice. They will avoid your apartment, because of the kitty.
I cannot say what my response would be to their wanting my financial records, because it would be ugly. I tend to hurt people's feelings when they really annoy me. Just ask our last landlord company, who decided to sit on our security deposit for more than 60 days (New York does not have a set period before it must be returned). Oh, the Attorney General's office might have some advice on them requiring you to provide financial information - that could be breaking the law. I would at least check and see if it way, because it is such a BS thing to ask.
I would not break the lease, but work through it the right way.
As for a new place, there are a good number of tools. Checking http://www.familywatchdog.us/
can usually help you determine if the area has a higher crime rate or people that might worry you (the site locates sex offenders living near an address). Also, check out the area at different times, including later at night and on weekends. If it is close, it shouldn't be a problem to cruise by a few times to see what it is like. If your state or county has a landlord/tenant assistance office, they can probably tell you if there are any complaints about the leasing person or company.
May is not a terrible time to be looking for a place. I would start my serious looking in March, if you plan to move in May.
15 years of moving a lot gives you some practice.