OP, I think your mechanic had been honest with you but there had been miscommunication/misunderstanding.
Your car has/had multiple issues looks like.
The squeaking noise can be very well due to the crank case ventilation valve that is integrated onto your valve cover on your car. The easiest way to diagnose this, if you ever catch it making this sound when you are parked, open the hood and try to open up the engine oil fil cap while the engine is running. If it is very difficult to open it due to a lot of vacuum and noise goes away after you open the oil fill, the noise is caused by the crank case valve and being part of the valve cover requires valve cover replacement.
From what you wrote it looks like you have been having oil leak from your valve cover too. You may also be having oil leak from your oil filter housing gasket. If oil had been getting onto your serpentine belt, it can cause it to slip off and then gets sucked into the engine behind the crank pulley and you will be looking at a expensive engine fix. The tensioner pulley also deflects by age that adds into the belt to slip off. Your mechanic may have seen this and replaced the tensioner. The tensioner pulley bearings may also have been noisy already. This is a good thing the mechanic did, especially if your belt was oily and/or the tensioner pulley had started to deflect.
The code P0012 code, as already have been written by other replies, can be the vanos selonoids, worn out cam ledge and seals. But it can be oil pressure loss due to the oil filter insert missing or has broken off piece. Easy to check too, open the oil filter housing cap where the filter is, check for the insert.
Also your Dad's mechanic may have just cleaned the vanos solenoid(s), not replaced them. It has been reported on this site that the cleaning of the selonoids are temporariy fix, problem comes back again.
See this TIS for more info:
https://static.nhtsa.gov/odi/tsbs/20...32779-8218.pdf
If I were you, initial thing I would do would be to clean the vanos solenoids, see if it will get rid of the code and shut off issue. If so, buy new solenoids and replace.
As to sludge, your dad's mechanic may have seen dirty vanos solenoid and concluded oil sludge issue. To some level he maybe right, but not necessarily. This vanos solenoid issue is so common. At the same time make sure you use BMW LL-01 speced engine oil especially if you are following the computer given oil change intervals. You may have been already doing this, just wanted to mention in case.