My questions:
1. Can she by herself go back to Canada/China and come back to the US without me accompanying her once her I-539 is approved?
-Yes. But she needs to have your new H-1 approval notice as she did for the first time. If she needs to travel, she does not need to waste money to file I-539 at all unless she will leave after July 31, 2018.
2. I believe she still does not need to get her H4 visa at the consulate, instead she can bring the approval of my new H1B (I-797 A or B?) and the approval of her I-539 to some POE to get a stamp. Do I understand the procedure correctly?-Yes, but she does not need I-539 approval notice.
3. My employer is a state university, and as a result they are not allowed to cover the expenses incurred for my wife’s H4 extension. So my school should not (could not) file my wife’s I-539 together with my H1B renewal request, instead my wife should file this I-539 for herself. Do I understand it correctly?
-Yes. But she needs to provide the copy of your H-1 extension receipt notice if she files I-539.
Thank you very much for your time and help!