Bmw325i-dontknow

Hi guys, I really need some help/advice/a new car! So last April (as in 2020) I bought a 325i on a 57 plate. Body type is e93 (not sure how much of a difference that makes in terms of replies). It developed the rough idle/misfire cold start problem associated with the following error codes: P0171 - System too lean, bank 1 P0174 - System too lean, bank 2 P2190 - System too rich at idle, bank 2 There is also a 4th code that comes up in reference to a misfire on cylinder 4 but I can't remember the associated code number for that one. Other abnormal things the car is doing and possible symptoms are as follows: When cold starting, if I gently but steadily press the Throttle, the revs are not steady but instead raise and lower at a uniform speed within a 500rpm range at any speed below 1500rpm. Sometimes the revs, when idling, drop so low (below 500rpm) that the car sounds as though it is about to There is a very small oil leak somewhere. The car has required a top up of around a litre of oil approximately every six weeks. The chrome tailpipe covers are black with soot. So here's what I've read as possible causes (if anyone can offer any further causes id be very grateful). A leak in the vacuum system. Not much more to say on this one other than that I couldn't see any obvious sources but wasn't 100% sure where I should be looking. A faulty purge valve. I read that "topping up" when filling the tank (pulling the trigger on the pump after auto cut-off stops flow) can force liquid petrol into the valve which is only designed to handle petrol vapour. Faulty coil pack. Pretty sure this isn't the cause as I've had all packs replaced once and the pack on cylinder 4 replaced a second time. Faulty spark plug. Again pretty sure that this isn't the cause as these have been replaced twice. Faulty injector. Not sure how much if any effect it would have on this but I have used fuel from supermarket pumps a few times until I heard it could potentially cause this. Is this true or just the stuff of urban legend? I have since only been using fuel from Shell and BP filling stations. I've also started adding redex to my fuel. Valves in need of walnut blasting. Throttle plate in need of cleaning. These last two are a little bit out of my depth. The place where I bought my car and have the vast majority of services, mots and repairs done have had a look at it a few times but can't find the cause and aren't bmw specialists so I am taking it to the local dealership for a full bmw diagnosis but this, of course, becomes a very expensive route to go down with them wanting £180 just to do the diagnostic test. I think that's all of the information I have to offer though one of you may think of something relevant that I have overlooked. I'd be incredibly grateful for any help or advice any of you might be able to give. Thanks Kerry