OK, it looks like the left door handle piece is part # 51416983703 but it's always good practice to double check this with the dealer you buy the part from.

Each of these parts are very easy to change out. The E60 5 series cars have been around for 10 years now so you can be sure that someone has done it before and probably posted a 'how to DIY' or even a video.

Here's a video for how to remove the airbag I saved to my favorites. Once you the the airbag out it's all self explanatory to remove the trim piece.
To remove the door handle trim piece you will need to take off the door panel. It's actually pretty simple to do. Two screws and a bunch of clips.
https://5series.net/forums/diy-do-yo...removal-29597/
https://5series.net/forums/diy-do-yo...h-trim-118185/
http://www.bavariansoundwerks.com/pr...dan_Logic7.pdf

 

Wow! Thank you very much for taking the time! I may actually do that soon. By the way, that's not my steering wheel in the video and that's also not my door in the other link. Are they so similar that the instructions will work just the same?

 

No problem at all...it only took a minute to look up those threads. And yes, the instructions are the same for your door panel and airbag.