Is there smog in Putian? Is transport good? As in the fast train to areas?

According to the history data, there is no smog during that time. It will be rainy in the most day at that time and the temperature will be 70F to 95F. You may take some T-shirts, pants, shorts, and thin jackets with you.
As I know, the sports center in the city is located at no.643, Middle Zhenhai Road. You may go there for the detailed information of the daily basis.
If you want to visit Beijing Great Wall, you may need to take a train which departs at 08:21, 09:39 and 12:00 every day. The duration is about 10 hours.
Binhaiwan Resort on Meizhou Island will be a nice choice. It is quite near the Mazu Temple. The transportation is quite convenient around this hotel. It charges CNY200-300/night for a double room.
In addition, it is quite moist in this city. You are advised to notice the weather condition before you go out. Generally, the typhoon will be frequent during July to September, which may influence your itinerary.