Perform leak-down test of fuel pressure; however, if the injector was weeping and bleeding down, you'd also see extra cranking on a cold start. The raw fuel smell is probably due to the extra cranking, not a leaking injector, but that's just guessing without data.
Check for DTCs (even hidden ones), look at your fuel trims, MAF flow rate and A/F & O2 sensor data. The hot start issue is probably due to the vacuum leak that every 173k mile car, especially BMW, is going to have; a cold start is open-loop with a purposefully enriched mixture and can mask an air leak, but a hot start would not see the same. If you disconnect the MAF before the hot-start, and the hot-start issue goes away, that is additional data pointing to an air leak.
All of the above diagnostics are free.
The IAC valve on an M54 is also a maintenance item, either cleaning or replacing.
Check for DTCs (even hidden ones), look at your fuel trims, MAF flow rate and A/F & O2 sensor data. The hot start issue is probably due to the vacuum leak that every 173k mile car, especially BMW, is going to have; a cold start is open-loop with a purposefully enriched mixture and can mask an air leak, but a hot start would not see the same. If you disconnect the MAF before the hot-start, and the hot-start issue goes away, that is additional data pointing to an air leak.
All of the above diagnostics are free.
The IAC valve on an M54 is also a maintenance item, either cleaning or replacing.