New GFCI outlet won't reset after tripping when load wires are connected
If I'm understanding, your circuit runs from

Service panel -> receptacle A -> receptacle B -> switches and receptacle C -> fridge D

In a perfect world, we try to use one GFCI device per circuit when it makes sense to protect the whole circuit.  In this case, it does not.  The farthest thing from the service panel is the fridge, and it's desirable to avoid as long as possible putting GFCI protection on a fridge. (I can't tell you to violate the law, but the requirement for fridge GFCI in new work implies a new fridge which is built expecting to be on GFCI.)**

I understand the lighting at C is also not playing nicely with GFCI. 

There is no way, given existing wiring, for a GFCI at A to provision GFCI-protected power to B and the C receptacle, but not the fridge. The wires are not there.

Given these constrictions, the most correct way to install this is with GFCI receptacles at each of A, B and C.   On each receptacle, you would pigtail to the LINE terminals only, and cover up the LOAD terminals.  The result of this is the lights at C and the refrigerator at D will not be on GFCI. 

To assure nobody tries to plug a second thing into the fridge receptacle at D, I would change it to a single outlet (just one socket).  

If at some point you want to repair the lights at C and move them to the LOAD terminals of the GFCI device at C, that would be fine.  Normally, when you don't have a fridge at the end of a string, you save a lot of money by learning to properly protect downline outlets with the LOAD feature. 



There is no reason to tolerate an "old fluorescent" anywhere.  First, find a T8 bulb that fits the fixture, then get a modern electronic ballast that is correct for the T8 bulb.  Change the ballast and enjoy state of the art, hum-free, flicker-free lighting.  Or you can defer it; code doesn't require lighting be GFCI.  



**  A GFCI trip on a fridge results in spoiled food.  Bad, especially if not detected by the food preparer - and before anyone tells me "the chef will detect this", baloney.  Try having a senior citizen with homecare aides coming in and out.  I assure you, food given to the senior is never tasted or even looked at closely. They will pour a bowl of cereal with sour milk and never know, and then coach the senior to eat it.  Seriously.