My eldest goes to Dolphin, the other two will follow, second in Sept 19. She was also a shy, slightly anxious child not helped by being very young for her year. She has come on in leaps and bounds since starting at Dolphin (she is now in year 1). We couldn't be happier with our choice of school. They have been wonderful with her from day 1 with building confidence and dealing with any little anxieties or problems. She is thriving, loves the school, has made friends and couldn't be happier.
The lack of outdoor space has never been a concern of ours. The children use the common twice a day at break time and in the lower school do an outdoor PE session there once a week (as well as one indoor one in the hall). The older children do plenty of sport and compete against other schools. There are also plenty of clubs. They also get out and about for school trips and to the local library and doing things like traffic surveys etc.
Re leavers' destinations the head is known to be very good and the school gives plenty of information and support re options (starting in year 4 with "Getting ready" evenings for parents). They try very hard to look at each child individually and what will be best for them, which you are really only going to find out gradually as they get older and progress through school.
Do get in contact directly if you want to ask more questions or chat further, it's a very personal choice but I can't recommend Dolphin highly enough.