BMW 320ci. Runs perfectly 1st start can run for 1000's miles no issues. 2nd start car takes longer to start,delayed throttle response,lacks power climbing hills prone to cutting out as revs drop.

i'd suggest getting the cylinder compressions checked , black smoke indicates burning oil so could be piston rings , also that would explain lack of throttle response and low revs

I would start by replacing the transmission fluid, that will likely resolve the issue.

two things: Pressure control solenoid first. Second issue is the valve body. These are common issues, but 9 times out of 10 its just the pressure control solenoid. Power steering, could have a leak, that causes air,which in turn causes noise or yes, a noisy pump.

sounds like a fuel problem. check fuel filter for clogging, it may be blocking the fuel flow. also check and clean air filter if need be. if that does not do it. try spraying and cleaning throttle body with carb cleaner especially at throttle plate. it may have carbon deposits around it! have a good day!!

YES IT CAN BE AIR IN THE SYSTEM CREATING FUEL STARVATION ...TRY BLEEDING THE FUEL SYSTEM.

check idle speed sensor on your intake manifold around 700 to 750 rpm is normal ,air temp may efect idle speed .

Change your fuel filter, and run a can of "Seafoam" in your next tank.....
Check your spark plug condition and connections, as well as your distributor cap

Sounds like you need gearbox oil (transmission oil in USA) .... check your levels and top it up.

I had the fuel pump recently replaced i just assumed the fuel filter was replaced also cause my fuel pump is in the gas tank. My air filter is clean and I have 4 fuel injectors, no carburetor. A friend of mine who has looked at my car for me says it sounds like there is some kind of air leak but can't figure out where it is coming from?

the computer is in relearn mode could take up to 20 miles