Those spindles were designed and marketed for sending back to the manufacturer or a specialist spindle rebuild company when they need rebuilding. They are not intended for owner repair.
All that said, you now have a grinder that has outlived it's support system, you think there might be issues with the spindle, and sending it to a specialty rebuilder can cost more, sometimes WAY more, than a good used surface grinder costs, if you can find anyone interested in taking the job on.
I don't know the answer to your questions. I do not like the oiling method for lubricating the spindle, it is just asking for grit in the bearings, may well already be damaged or ruined. I still do not know how the proper spindle oil level is maintained, or how any overflow is intended to be drained away other than possibly into the motor, by what you have discovered so far. That is not an acceptable oil overflow method, if it is indeed what is intended. You also do not know what oil was intended to be used in the spindle. The spindle is not original to the machine, so there is not mfg. support for it through B&S, Excello, or Pope.
I cannot imagine it was designed to be put together permanently, never to be disassembled again. The rebuild label from a famed rebuilder shouts otherwise. First, cause no harm. Look it over and find a way to take it apart _very carefully_. Note carefully the orientations of each inner and outer bearing race mountings, to the spindle and to the housing, also front and back faces as you remove them, and mark them carefully. Mark front and back bearings if they are not different. Clean everything spotlessly. Truly spotlessly. Then inspect the bearing balls and/or rollers for damage and for wear from the bearing races spinning in the spindle or housing. If the bearings show damage, see about replacing them and whatever seals are present. Be prepared for sticker shock for the bearings. If there are other wear items, see about replacing them. Make a decision on how to proceed from there...
That is how I would approach it, YMMV...