Documents all the problems in writing and make sure you get proof that the management company received your letter. Be sure to review the lease very carefully and make sure that you comply with all notice requirements in the lease.
You may want to contact your local housing department regarding the rodent and heat issues.
Whitcomb Law Firm, Ltd.
Illinois law allows you to have some of these repairs made and maintenance issues addressed and deduct some of the costs from your rent, and then Illinois law is supposed to prevent the landlord from evicting you for doing so. That could be a fight but it is your home. Check out 765 ILCS 742/5 - Illinois Residential Tenants Right to Repair Act and 765 ILCS 720/1 - Retaliatory Eviction Act. However there are limitations and neither allows the tenant to "break" a lease. What you are trying to do is assert a "constructive eviction" as well as illegal eviction (changing of locks). It would be worth your while to consult with a local attorney to decide on a course of action. You MAY be able to break your lease but this forum just won't go far enough to make sure you are protected -- a face to face with an attorney would be the best thing to make sure all issues are properly addressed.