Easy Question
 

B1 is obstructed at 1B, misses 1B, is safe at 2B. Time is called, defense appeals that she missed 1B. Is there any way for the OBS to occur where she is not out in the situation where she continues to 2B?

Generic easy answer is that BR is out.

Your second question is more complicated, and will generate debate. Is there ANY case where an umpire would not call her out. Probably - but it would have to be blatant, intentional obs that I don't think I've ever seen. Hypothetically, if F3 set up in front of first base, waited for BR and pushed her toward 2nd, perhaps even preventing BR from trying to touch 1st a 2nd time, I can see an umpire extending his authority to disallow an out at 1st base. (And I can see one going strictly by the book and calling her out anyway - I suspect this one would be 50/50)

But generally, BR must still touch bases in the proper order, even if OBS'd - this is one of the exceptions to the "cannot be put out between the bases where she's obstructed" part of the rule.