Date registered: Jul 2006
Vehicle: 1999 E55, 2001 E320, 1990 560 SEC, 2004 ML350, 2001 ML55, 1995 S500, 1998 SL500, 2010 E550 Sport zoo
Location: NW Arkansas, La Jolla CA, Hong Kong
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To be frank, honestly you should not be hearing a "buzz".
1) I would choose to replace the fuel filter first, and note this is a part that has both China non-name knock-off's and the correct OE part. The correct OE part should be used, the China knock-off's have very short life as compared to the original OE part.
If the new fuel filter does not solve this issue, then you can look later to replace the fuel pump and the same China-knockoff-no, OE part-yes rules apply.
2) The window issue might be a battery if the car was not running, but most commonly when this problem is reported it when the car is running and that most often is a switch problem, and the center console switch plate is one mechanism which is only available as a OE part.
If you are suspecting your battery for other reasons, then best to replace the battery.. normally if you suspect the battery, you do want to replace it, since once it really "dies" you will be dead-in-the-water on the road (or in you case at a parking garage) which is the least desirable thing. you have a ton of auto batteries in HK, best to make sure you get one (standard capacity OK) that fits correctly under the back seat and importantly has the proper vent tube connection that will match the existing MB battery vent tube to vent the battery fumes down and outside the car to prevent future corrosion in that area.
Keep the beat !