Is it possible to run a windows OS off a Flash drive/thumb drive on a Mac?

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.

Yes, is the short answer. Here's a guide for 8.1 but the same remains true for W10. The only difference here is that you would need to partition the USB drive s it's bootable on startup.

http://www.intowindows.com/how-to-install-windows-8-1-on-mac-without-boot-camp/