Any idea what a good rate is for a 2 bedroom 1 bath with family room (no kitchen, but a mini fridge and microwave) should go for, and how often are they typically occupied?

The fridge was one of the first things requested in the contract.

The renters are paying for the room and some shelf space in the kitchen/fridge/bathroom.

I'm looking at investment houses and some of them list the owner is responsible for fridge, stove and microwave.

Charge her prorated cost for fridge she removed and cost of removal.Give her notice your coming in for repairs, per city inspection, and if she doesn't let you in call city inspector. 

If it was legitimately an issue with the fridge, I'd offer to pay for it. 

If renting out bedrooms, a small kitchenette goes a long way (mini-fridge, toaster, microwave, coffeemaker with good coffee, etc.).

My concerns: No previous rental data on MLS for that unit, current tenant is occupying only upstairs part of the unit (with no kitchen, has his own fridge), As per seller's agent paying $500 and said he may leave at request.  

When I purchased the property a washer, dryer, and fridge were included in the deal.

In your case, it is, which is the fridge.