The temperature at that time is favorable. Not hot, because it is only average 28C in its hottest month, but it is rainy in May and June. For your questions:
1. Airport bus to the railway station of Guilin, 20 yuan per person for 30 minutes' drive, or 80 yuan for taxi which can take you just to your hotel. Sightseeing places, you can take it for reference
https://www.travelchinaguide.com/attraction/guangxi/guilin/
2. Guilin Bus Station has direct buses to Yangshuo and it will take 1 hour and 20 minutes. The better idea is to have a river cruise from Guilin to Yangshuo along the Li River.
3. The river show is at the Menghuan Theater at 95, Qixing Rd, Guilin City. It begins at 8pm at night and the fare is about 120yuan.
4. Bamboo raft is take to cruise the Yulong River. There are cruise ship from Guilin to Yangshuo along the Li River. Suggest fo no take rafting, expensive and the scenery is not food.
5. Guiling has express buses to Longsheng (1h40m)and then change to another bus to Ping'an Village (1 hour). There are also ordinary bus running directly to Ping'an Village. Vice versa.
6. Suggest you take train from Guilin to Guangzhou. T40 (9:28pm-8:38am), total 11 hours. Coach is not comfortable.
7. When you come back you can take T38 (7:47pm-6:42am). the ticket fare is about 320 for soft sleeper and 100 yuan less than the hard sleeper.