I don’t have the time or inclination to do your research for you, but I do know that living in or near Princeton will cost you. A university town (particularly an Ivy League one) requires deep pockets to live.

Talk to a realtor.

Yes, apartments come with stoves and fridges and usually dish washers (often not washers and dryers…which may be the basement)

Jackson and Howell will be quite a bit cheaper for you than the Red Bank/Colts Neck area. You could look in and around Freehold as well; there are some cheap apartment complexes (Chesterfield Gardens) along Rte. 537. My husband paid $900 for a winter rental apartment in Belmar this winter and is now paying $1,000/month in a retirement community. In Ocean Twp., Monmouth County you could try Continental Gardens or Middlebrook at Monmouth. Pleasure Bay Apartments in Long Branch is quite nice and a friend is paying $900 per month for a large studio there.

And yes, fridges and stoves will be included.

One other consideration is your commute… where do you work or go to school?
As @gailcalled mentioned, Princeton is very pricey, but some of the outer areas aren’t too bad. I’ve lived all but the first five months of my life in the area you are referencing, so once I have an idea of where you will be working and studying, it will help me narrow it down for you.

Most apartments do have major appliances but are not furnished. Many also include gas heat and hot water, but you would have to pay your electric bill and cable TV.