Post some pics of your upcoming adventures. Sounds like you have some fun in your future...
The mpg gauge isn't getting a mph signal, definitely. Since you'll be taking the cluster apart to R&R the SI board, maybe check the pins that connect it to the board. The gauge will slowly bounce from 0 to 40 if it isn't receiving the mph signal from the diff sender. The driver board filters that signal, then sends it to the speedo, the economy gauge and also provides three outputs with the green snap-in adapter installed in models with cruise control.
I'd figure the hypoid is what you want, but I'd assume all gear oil is pretty similar. Hypoid gears are cut in a curved pattern for lower noise, as opposed to straight-cut gears. I'm unsure of what they're talking about, as hypoid diffs are virtually universal in cars. Look on the side of the tranny for a label. BMW spec'ed three types of lubricant for the Getrag 240. I'm not sure why...but I've tried both gear oil (Valvoline synthetic 75w140) and ATF (Valvoline DEXRON II?). The ATF definitely improved shifting in mine.
I've also heard good things about Redline oils, Amsoil (if you can get it reasonably), and an organic alternative I'll be trying soon, RLI in Cleveland, OH.
About the SI board...there are two models, be sure you're ordering batteries for the correct one. Earlier ones had NiCd rechargeables, later ones had lithium non-rechargeables. The two boards are interchangeable in all clusters, but the batteries are not. IIRC, early batteries should test about 3vdc combined, the later batteries should each test out to about 3.7vdc each...if they're still good.
If you're really careful, the cluster can come out with the steering wheel still installed. Though I have replaced the airbag wheel in my car with a 'vert leather sport wheel, so YMMV. If you do decide to remove the wheel, be sure to disconnect the battery so you don't put the airbag light on.