Stay out of drama. Just do your job and do it well. Be polite. If she lies about you, approach your manager and correct the record. Be mature and diplomatic.
I would keep written notes of dates and times of things that have happened. You’ll know when you can’t take it anymore and have to bring it up to your superior. Definitely make sure you have established with her in a professional manner, that what’s she’s doing l/saying is not ok with you! That way she can never say that you didn’t say anything to her about it or that she thought it was ok to do.
She's the manager's sidekick. The manager actually employed her husband for a short while
If it's not your job to be training then you need to say "hey sorry I need to focus on working right now and it's a little distracting when you're next to me, can we chat later?" And if she cries, not your problem to manage other people's reactions. You can also tell hr what you're experiencing if it becomes a real issue.