Yes (and they will also check it if you are flying from Taiwan to China.)
Also she will need a visa like stamp in the book. In my experience it has also always been single entry and took 15 days in the place of household registration each time. This means that every visit to China has to include 15 days in the hometown. It also means that you cant transfer flights in China, which reduces travel options.
Again it was in 2012 but the first time my wife landed in TW it was through HK and she had no pink book. She returned to mainland about a year after, through a direct flight and applied for it afterward.
Yea, we did the exact same, and got an earful on the way back when we landed, hauled into a side room, made to wait for hours, and a permanent flag on the electronic record. That was 2014.
gosh if you make that sort of money to pay tax to the tune of usd 1300 per day. you should be able to find some good attorneys that are savvy with the tax laws both for the USA and Taiwan.