I'll bet that the battery & alternator are OK. It's great that you're getting them tested though, you're eliminating causes one by one. I strongly suspect you have a pretty severe electrical problem to chase down. Granted, they're a pain in the ass, but they're much cheaper than throwing new parts at the car. You'll also get to know that car inside & out doing stuff like this.
If you have a severe short it'll give you those symptoms. The car can't run on the alternator alone unless the electrical system is in excellent shape. I'd guess that the car will set a CEL if voltage is too low...many things in the car must have a little over 12VDC to run properly, especially injectors, coils, etc. All modern ECUs have a voltage correction circuit built-in to handle a discharged battery.
So fire up ye olde old digital multimeter... If you don't have a decent one, I'd highly recommend it. Sears & Radio Shack sell economical ones. You won't need a Fluke. Alternatively, go get two of those $4.99 cheapies at Harbor Freight (they only live a year or two).
My first test is current draw...I charge the battery (it should test to ~12VDC unplugged), install it in the car. Switch the DMM on amps & install it inline right in the middle of the battery - terminal and the -wire. You could use the + side but you have to be careful not to short it on the frame of the car. See how many amps the car is pulling at rest (don't even put the key in yet) maybe 30 seconds after you reconnect the battery...mine usually is about 200-400 mA. Should be less than 1/2 amp (500mA). I then put the key in, turn it to pos II (no fan or stereo) & see how much the car is pulling then. IIRC shouldn't be much more than an amp or two.
So if the car is pulling more than 1/2 amp, get some helpers (kids work great) and have them look for the current to drop while you remove & replace the fuses one by one. When the current drops, you'll have your circuit; then look that circuit up in the ETM & test it out. At that point I'll just leave the fuse out & drive the car a while to see if it's fixed. Keep in mind that there is also an unfused side to the car...the starter, DME and fuse box itself aren't fused so if the "pulling fuses" test didn't do anything your short is likely there. Also keep in mind that early e36's are famous for flooding the DME & pinching brake light wires in the trunk hinge area.