i have a 94 EX accord, 4cyl, 148k on rebuilt engine

so my car is having a problem starting, went to buy a battery and they said it isnt the battery or altenator, those are still good, when i try to start it, it turns but doesnt start, but when it does start i hear a strange grind noise right when it starts

and my radiator fan runs when i turn off the car, is this normal?

 

Addressing the start problem, check for spark to the plugs when cranking, and listen for fuel pump to run 3-4 seconds when the key is cycled on.
When the engine is shut off Temp sensor 'B' [in line with top Rad hose] will let the fans continue to run if the head temp is high.
When the sensor goes bad it lies to the fan control unit.
A simple check is to unplug the sensor and see if the problem goes away.