Yes once you have an ARC you can sponsor her entry permit. It is unlikely but possible that your school will do that for you. However you can do it yourself with your ARC. The immigration officials in Taiwan are used to the process and in my own experience have been very very helpful and nice about it. However be aware that there are document requirements. This will include your marriage cert - translated and notarized. Best to do that part before you leave: ask in TECO in your country.
If you were married in China you have a trickier process where you have to get it notarized by the cross straights bodies, and that would involve travel to China. Also be aware that you may have to arrive first, get your arc and then apply for her permit, so she may need to follow a month or two after you arrive.
Take a look at this thread: Taiwan visa for a PRC spouse
Based on your ARC and marriage cert only yes. If she has qualifications she might be able to apply for her own work permit (but I have no knowledge or experience about that). Also be aware that unlike spouses of ROC citizens she will not be eligible to apply for a TARC and eventual ROC citizenship after spending x number of years in Taiwan (as far as I know, that is what the immigration officials have told me). If you get an APRC her entry permit is set to a longer duration (3 years).
Be aware that if you wife is travelling directly to China from Taiwan and does not have the Chinese pink entry book + permit stamp she will get a roasting from the Chinese immigration officials, and it is imaginable that they could escalate this to a freeze on her passport. Best to travel to China indirectly if she doesn’t have the documents, she could then apply for them while in China. She would need your ARC and marriage cert to do that, and it can only be done in the place where her household is registered. (She cannot board a flight from China to Taiwan without those permits.)
Also some - not many - check in staff in airlines are not familiar with ROC entry permits and may question it when travelling to Taiwan from third countries. Some check in staff in some airlines just issue the ticket without even asking for the entry permit, so experiences vary a lot there.
It will be most likely be a easier for her to settle in than for a non Chinese person. Most all Taiwanese are entirely non discriminatory towards PRC citizens. There are some windups, be ready for dalu dala dalu on the news, but overall the problems are really a lot less than you might imagine.
Best of luck and feel free to PM if you have more detailed questions