I just changed my outer covers last week, so I think I can help here.
"Is this black box with pins part of the cover or is it a separate part to order?"
The black box is a separate part independent of the cover. Personally, I would try the plug first before even considering ordering the black electrical box. Electronics in our cars are insanely expensive, and the cost for that part may very well be close to the cost for a used driver's seat that would come with that part, and more.
Which color would you get? Is it "grau"?
You would get grau, yes. However, notice that your seat cover is scratched and the white plastic is showing through. My grau seat covers were in a similar condition and fixing the cosmetic defect was the genesis of my replacement effort. I would suggest getting schwarz (black, BMW part #52 10 7 058 008), because black will hide the future scratches that are inevitable. This is possible because the black seat covers are made of black ABS plastic and not painted, thus a scratch will not take off paint and expose an off-color base material.
Some tips, since you asked:
- The seat control buttons simply pop off, but are held on very securely. It's nerve wracking, but they're just on there tight.
- The seat covers themselves are each held on with two Torx screws and three plastic rivets. Again, the plastic rivets are in very tight and will likely necessitate destruction for removal. These rivets create a lot of frustration. Replacement plastic rivets are sold in sets of six (BMW part#52 10 8 150 020) and are markedly inexpensive, so I would urge you to get these as well.
- The seat covers are sold in a set of two, and come with one out of six holes filled in with a button cover. Since you have the "basic" electric seat like I do, your controls will fill three more holes, so if you wish to change colors you'll need to purchase the final two button covers, which curiously enough come with two each of the button covers that already come with the seat covers. If you purchase the seat covers and the button covers, you'll end up with two extra button covers per side. Now I'm not sure if this was an idiosyncrasy with my dealer, but a set of seat covers came with two - one for the driver and one for the passenger - while a set of button covers was for only one side. I needed to order an additional set of button covers. To summarize, I purchased one set of seat covers and two sets of button covers. This will fill in all of the holes with four identical button covers to spare. I have leftover buttons in both grau and schwarz, as well as the passenger's side seat cover. If you wish I can sell them to you for a dollar or two over what it will cost me to ship them.
To remove the rivets and the control buttons, I would highly recommend interior trim panel tools. I acquired the first two of the following three sets from Harbor Freight Tools for about five dollars each. The set of yellow tools is particularly durable and desirable and has proven to be worth much more than the asking price to me:

If you need more information, please see the following thread in which these seat covers are discussed ad nauseum:
http://forums.bimmerforums.com/forum/showthread.php?t=842536
"Is this black box with pins part of the cover or is it a separate part to order?"
The black box is a separate part independent of the cover. Personally, I would try the plug first before even considering ordering the black electrical box. Electronics in our cars are insanely expensive, and the cost for that part may very well be close to the cost for a used driver's seat that would come with that part, and more.
Which color would you get? Is it "grau"?
You would get grau, yes. However, notice that your seat cover is scratched and the white plastic is showing through. My grau seat covers were in a similar condition and fixing the cosmetic defect was the genesis of my replacement effort. I would suggest getting schwarz (black, BMW part #52 10 7 058 008), because black will hide the future scratches that are inevitable. This is possible because the black seat covers are made of black ABS plastic and not painted, thus a scratch will not take off paint and expose an off-color base material.
Some tips, since you asked:
- The seat control buttons simply pop off, but are held on very securely. It's nerve wracking, but they're just on there tight.
- The seat covers themselves are each held on with two Torx screws and three plastic rivets. Again, the plastic rivets are in very tight and will likely necessitate destruction for removal. These rivets create a lot of frustration. Replacement plastic rivets are sold in sets of six (BMW part#52 10 8 150 020) and are markedly inexpensive, so I would urge you to get these as well.
- The seat covers are sold in a set of two, and come with one out of six holes filled in with a button cover. Since you have the "basic" electric seat like I do, your controls will fill three more holes, so if you wish to change colors you'll need to purchase the final two button covers, which curiously enough come with two each of the button covers that already come with the seat covers. If you purchase the seat covers and the button covers, you'll end up with two extra button covers per side. Now I'm not sure if this was an idiosyncrasy with my dealer, but a set of seat covers came with two - one for the driver and one for the passenger - while a set of button covers was for only one side. I needed to order an additional set of button covers. To summarize, I purchased one set of seat covers and two sets of button covers. This will fill in all of the holes with four identical button covers to spare. I have leftover buttons in both grau and schwarz, as well as the passenger's side seat cover. If you wish I can sell them to you for a dollar or two over what it will cost me to ship them.
To remove the rivets and the control buttons, I would highly recommend interior trim panel tools. I acquired the first two of the following three sets from Harbor Freight Tools for about five dollars each. The set of yellow tools is particularly durable and desirable and has proven to be worth much more than the asking price to me:



If you need more information, please see the following thread in which these seat covers are discussed ad nauseum:
http://forums.bimmerforums.com/forum/showthread.php?t=842536