Can a process run if there's not enough space for minimum pages?
There's no set answer; it will come down to what the OS is implemented to do in that situation.  I suppose if you implement the OS poorly, arbitrarily bad things could happen.

But in practice, with modern OS's, there's no such thing as "minimum pages the process needs".  The way virtual memory works is that the process has no idea what pages are actually mapped into physical memory at any times and which ones aren't; the OS transparently pages in pages as needed.  So, that notion doesn't exist.

Thrashing is not a black-and-white phenomenom.  For instance, thrashing can happen if 98% of the CPU time is spend paging pages in and out of memory, and only 2% on running the process itself.  In that case, forward progress is still being made -- just very slowly.