Dang it, I missed a post on keichii's bike! since you were researching for a fanfic KeichiiK1, here's my take on the subject.
Identification of Keichii's bmw is made difficult by several factors: the sidecar, which often hides most of the identifying features, the replacement/custom parts which either aren't bmw or are from a different model, and the rarity of this particular bike. I am not going to say much about the sidecar, as they are often custom built and of little use in identification. The "custom" or replacement parts include the fork assembly, the front wheel, the front fairing/headlight, and the seat.... generally the most obvious portions visible on the bike with that sidecar in the way.
The fork MAY be a rebuild of an r68
http://www.bmwmc.dk/images/_models/r68.jpg 
, or scavanged from another brand bike, but the placement of the axle at the very end, rather than the top side of the end of the fork, is somewhat unusual. The front wheel has an incredibly large drum brake, pre 1973, most likley candidate is the R50
http://www.bmwmc.dk/images/_models/r50.jpg 
or R69s
http://www.bmwmc.dk/images/_models/R69S.jpg 
. The front fairing is very much racer styled -probably custom, but I did see a Honda fairing/light/windshield that fit the bill perfectly in all but color... sadly i can't find the pic. The round signal lights are also probably non-bmw, unusal because they're round and installed INSIDE the fairing.The seat is also in the style of a racer, such a solo seat is almost certainly custom built
http://www.suicidemission.com/slash5.htm 
.
The fuel tank and engine are a different story.The huge almost rectangular tank, and the unusual design of the boxer engine's cylinders marks this as a 1954 "rennsport" (or racing)
http://www.bmwmc.dk/images/Walls_3/BMW-Rennsport-1954%20text.jpg 
here's a model kit:
http://www.seinando.com/kims%20hd&bmw.JPG 
. The four bolt pattern of the valve cover with a single shaft for the valve cam is, as far as I can tell, unique to that engine. Since the racer was produced in only the one year, and in extremly limited quantity, it is no wonder all us AMG fans have such a hard time identifying it. With such an odd mix of parts, and given the racing nature of the motorcycle, I'd say that Keichii (or possibly his father) bought the bike after an accident without the front end, and rebuilt the bike as a racer, and as keichiik1 said earlier, added the sidecar when he went to university.
the 1954 BMW Rennsport, had a displacement of 494 cc, and developed 48 hp pushing it to speeds of about 130 miles per hour, 210 kph.