How to safely wire fluorescent lighting with three prong plug on the ceiling?
Extension cords are not supposed to be used for permanent fixtures. These fixtures are supposed to be hard wired or directly plugged into a permanent outlet.

A common approach to solve this problem would be to install a switched outlet on the ceiling near the fluorescent lights. In many jurisdictions, you could use non-metallic cable to run the outlet and the switch. In some, you might have to either snake the cables in the wall or use surface raceways.

You could ask your landlord to make this change. Good luck.

Reducing the number of extension connections would somewhat improve safety, and ensuring that cords are not dangling in walking space or prone to being pulled or tripped over would reduce risk. One extension cord to the fixture is better than a cord and a power strip. Also make sure that any extension cord is three wire. DO NOT BYPASS THE GROUND PIN! But any of these patches are less than what is considered safe practice.

At very least, you may want to install a GFCI outlet wherever the cords are ultimately plugged in. This would provide some protection in the event of failure of the fixture or the cords.