starter and ignition question

[nextelbuddy]

my new to me 4.8is had the fuel pump replaced 2 years ago for $650ish dollars...it died on me a few days after taking ownership. I removed it and found that they did not use an OEM fuel pump, it was a generic 3rd party pump that had no markings on the pump itself. literally a no name brand pump and sending unit for $600+ dollars what? lol

ordered a replacement pump and sending unit for $305 from fcp euro but now the car has an issue where on cold starts it starts up perfectly and of the engine is warm/hot then it takes a LONG time to start.

my issue also is i dont have to hold the key in ignition III to keep the starter going, the starter seems to run on its own until the engine finally starts or it gives up after X amount of tries until i try again.

i will be addressing the fuel filter soon (waiting on the part from FCP Euro) and that should resolve the long start issue

but is the starter supposed to keep running on its own until it detects the engine has caught and started?

none of my e46 ever did this so if its normal then its new to me lol