Is it possible to run a windows OS off a Flash drive/thumb drive on a Mac? In theory, is it possible to run Windows OS off a flash drive, so that on starting up the machine, if the stick was in the USB slot, it would startup on that 'hard disk' and run windows programs off it?  

I'm basically asking because I'm having nightmarish issues trying to get Bootcamp to partition my drive and run Windows on the partition. I've been searching the problem and it seems to be an ongoing problem going back to 2014, which amazes me.
In theory, yes, but in practice it would be quite slow and cumbersome doing so from a flash drive. 

My suggestion would be to search here for an answer to whatever your Bootcamp issue is (or ask a question about it). 

However, if you really want to bypass the internal drive, then I would recommend using an external hard drive (instead of a flash drive) to run Windows from. Be aware though that the process for doing this is quite involved. For example, read this question for one way to achieve what you want.