I just acquired this bike with an engine that has been torn down a bit with the oil pump removed and completely disassembled.  I'm attaching a picture and don't know yet if I actually have all the pieces.  My question is if I can simply keep it entirely removed and go ahead and run pre-mix - or does the assembled pump still have to stay in place for some reason?

Best practice is to remove the pump driveshaft and buy or make a block-off plate for the hole in the clutch cover.  Driveshaft bushing is lubed by autolube pump oil.  Clutch cover may need to be removed to remove pump driveshaft. 

Then you can premix.  If you don't remove the shaft, it can seize...  not good.  That said, if you just want to start the bike and run it a little, it won't hurt to leave the driveshaft in place temporarily without a pump...  I just wouldn't do it long term and maybe try to oil the bushing a little. 

ebay block off plate

I like running oil pumps. 

asco

If you’re going to go to all the trouble to remove the clutch side cover, drain the oil, put a new side cover gasket on, after thorough cleaning, it makes sense to me, if you already have the drive gear in place that runs off the primary gear, to get yourself a pump and connect it. Along with the splitter for the throttle cable on the oil pump, a couple of banjo fittings. And, if you’re really concerned that the pump might fail, add some two-stroke oil to your gas tank, which will prevent rust as a back up.

Asco

I love the yamalube pumps, both at the gas station and how bike runs.
its a very clever device that works great, prefer over premix