Stevo; It appears the battery is OK, and from he voltage shown when running, the alternator is working. But as already stated, these are fussy cars, with very sophisticated electronics, and yours is telling you it's not up to snuff. It's time to throw in the towel, and either bring it to a dealer or a good specialty shop that has some better diagnostic equipment. If you're looking for an independent shop, the very first question should be: 'Do you have an Autologic with up to date BMW Software". If the answer is no, move on. Once they sort the battery light, ask them to see if the Autologic can communicate with the ABS module. If the answer is no, tell them to NOT fix it. If it's a bad sensor, that's up to you, but likely not crazy expensive.

Either the ABS problems will solve themselves when all voltage issues are fixed, or they won't. If they don't, often the module is completely out to lunch. While it could be a sensor, if its the module, you can either: Put in a new part, then have it coded, ($1,500 part.). Put in a used part, and have it coded, ($750 part). Or send your module out to one of the rebuilders on eBay. If you're lucky, $100, and then recode. If you're not lucky, another $400 and they'll send you a rebuilt unit. No matter what, the unit will need to be coded. But to attempt this with a battery light on is silly; these cars are known to throw the trifecta lights when voltage is not perfect. Sort that first.

The ABS module in the X5 is in a very bad spot. It's about a minute to remove from a 5 series, but on the X5, you need to remove the left side headlight, and the whole assembly with the oil filter, power steering reservoir needs to be moved aside. Given the location down below the headlight, mine was completely filled with muck...no wonder it failed.

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2014 BMW 328i Xdrive
2011 BMW 335i M-Sport
2008 BMW X5 4.8i Sport
2000 BMW 528i 5sp