How do I start a case against a potentially-infested apartment?

Mold is a complicated issue because many molds are common indoors and out, and are harmless. You need to find an expert who can test the type of mold that exists in your unit and tell you whether it is a harmful type. Try a Google search for mold-testing.

If the mold is harmful and makes the premises uninhabitable, that could be a defense to your obligation to pay rent. You need an attorney who regularly represents residential tenants to help you with the real estate side of this.

If you have been sickended by the mold, you may have a case against the property owner. For such a case, you want an attorney who handles personal injury cases.

As for the refrigerator, that is inexcusable. If the refrigerator is included in the lease, then the landlord breached the lease and is responsble for damages, including the cost of food. You may also have a case for the rental value of the refrigerator. You are entitled by law to 24 hours' notice before the lessor may enter your unit (unless there is an emergency).

I always tell potential clients that if they want to pursue a mold case, they need three things:
(1) a report by a very good industrial hygienist confirming the presence of toxic mold--these are expensive
(2) an illness
(3) a doctor who will say, under oath, that THAT mold caused THAT illness.

Without those three things, you cannot make a case. Even with those three things, mold cases are extremely difficult to pursue. Generally, insurance policies have exclusions for losses caused by mold, so the landlord would end up paying any judgment personally. This tends to make people fight harder.

Two weeks of rent is not a lot to give up for your health. I would advise moving out and then suing the landlord for any money he withholds because you moved out early.