Once you have the GC, wandering across the border is simpler. Make sure to register for echannel (usually somewhere just after China checks your passport heading in). Once you do that, your next trip in or out should be far simpler.
Make sure you are inside of China for at least 3 months each year. There's a way to get a special exemption if you have a pressing need to be out longer, but reports on how easy that is to get are mixed.
1. Beijing recently made it VERY clear that there are NO "foreigner" hotels, despite persistant beliefs in rules that ended over 20 years ago. ALL licensed hotels can accept foreigners (limited exceptions for areas where foreigners are forbidden to enter or require special permits). Note that this may not apply to AirBnB equivalents. If you encounter a hotel claiming it can't accept forigners, contact the police and also the National Immigration Association. In some cases, you may be forced to give up, but always DARE them to try to scan your green card into the hotel registration machine.
2. Depends. Does a bamboo raft on a river work for you? If so, Guangxi has multiple candidates. Do you prefer mountains, deserts, grasslands, forests, or something else?
3. Pick a province and use your favorite search engines to look for "Best tourist places in _____".
4. Xizhang is one area where you'll need the same special permits as other foreigners.
For a start: Great Wall (anywhere along it), Southern Wall (in FengHuang), Moutai town
, Xishuangbanna, Yangshou (take a raft to see the back of the 20 rmb bill).