Have to use copy and paste functions.
1.I don't know about any other cities. Both in Tianjin (my in-laws' hometown) and my hometown Guangzhou, police follow the same rules.
2. I paid $70 in Chinatown for the 寄养委托书. I wrote a statement and handed to the man whom I paid and he sent it to NY consulate to 公证. I think you can do it by yourself. I was just busy for other things at that time and wanted a shortcut.
3.I've never tried that before. But I guess it won't work. The travel cocument won't get a record of entering China, how could you possibly use that for departuring China? They do care about this. Otherwise, the U.S. passport without a entry record will be useful for your kid to depart from China. But actually it doesn't work that way.
What you mentioned in your post is just what my sister and I both experience in two different cities. Hope it will help.