There are nonstop flights from TPE to KWL on EVA but may not be daily...but if you could do an award all the way to KWL with stopover in Taiwan on the way, that would work pretty well. Then return from Shanghai or Hangzhou. Try very hard not to route through PEK at least on the way inbound. Jiuzhaigou may not be a place that you can get awards in/out of due to demand. And regular flight tickets are quite expensive. If you choose Huangshan (TXN), then flying from there to PEK and picking up an award PEK-home might work. Doubt if CA is offering awards that would include the TXN - PEK segment but you can try.

More confusing to me is your time frame. I thought you started with only 9-10 days for a trip. I think given your wish list, you may have some unrealistic time frames concerning intercity travel and particularly trying to get in/out of Zhangjiajie. Getting from Guilin to ZJJ is not a straightforward or time-efficient exercise. For an adequate trip to both including the transportation, figure on 7 days right there. Additional if you want to put days in Hangzhou and/or Shanghai. Jiuzhaigou is not in the same geographic area. It isn't all that high, and altitude is normally not a problem for most people. The upper reaches of nearby Huanglong (a separate park), does get quite high and that can be problem for some. But again, if you want to carve out time for that, at a practical minimum you'd need 3-4 extra days to do the plane connections then time at the park(s) themselves. Huangshan is quite accessible from Hangzhou by road and Shanghai by air, so one of the easier itineraries would be Guilin - fly Hangzhou - road to Huangshan - fly Shanghai. This could be done in a week. Make sure that you understand the setup at Huangshan--cable cars are available but a lot of stair work up and down will still be required. There are photogenic villages in the area that might be of interest to you.

I can't speak for Taiwan, but central and mountain China would typically be better in September than in early June due to weather (rain) issues. Try to avoid July, August (school holidays and all tourist sites exceptionally busy) and at all costs avoid the first week of October (national holiday).