Hello, this is my first post on this forum. I'm not new to horses but new to ponies and need help with a little situation.
We leased a green section B welsh last winter and my 9 year old daughter fell in love with him.
Horses & ponies are still all equids and behave similarly in similar situations. You leased, then bought, a green pony for your greener 9 year old daughter. Not something I would have recommended. The expression 'green on green equals black & blue' came about because it's a pretty true statement.
We always lunged him before riding because he was energetic in the cold weather.
Not at all unexpected, with the cool weather they have more energy for something other than sweating. However, I wouldn't rely on lunging too much or too often. You end up with a really fit animal who now has even more energy to act silly. Lunging before you ride for 5-10 mins each way, to give the horse time to 'get his brain in gear' isn't a bad idea, but any longer than that you're not really accomplishing anything.
2 Months into owning him and moving to a new barn, he unexpectedly bolted at a gallop while she was riding in the arena. She fell off and was not hurt thank goodness. Another rider cantered too close to them and he lost it. It took 3 weeks for him to settle again, he was really freaked out about her flying off him.
Not unexpected at all, the horse came barreling up behind him, whizzed past and may have made nasty face/ears as he did it and scared him silly. He's green and obviously hasn't learned that it's ok to be passed by another horse. That takes a few times of being passed and learning nothing bad will happen. Also not unexpected that he got freaked about your daughter falling off, it might have been the first time someone got unseated off of him and, again peripheral vision, he saw something flying near his eye and it shocked him.
This week she put her rain jacket on while mounted - I was holding on to him - and he bolted again. I was holding the reins but he dragged me a few feet before stopping. Daughter grabbed mane and stayed on.
Again, totally not unexpected. I have older BTDT horses and I would not try to put on a crinkly rain jacket or poncho while I was still mounted. They have to learn not to be frightened by things like this, it takes repetitions. For now, if she needs to make any changes in her clothing, hats, helmets, gloves, whatever, she needs to dismount and do it on the ground. Let the horse see the items, rub them on him, crinkle them up (in the case of a rubber raincoat or poncho) so he gets used to the various noises.
Another situation for "Watch Out": I would not hand her a cup with ice or a sports bottle with anything in it while mounted. That noise can surprise him and get a BIG reaction. One of my sisters worst falls ever was because she took a sports bottle with ice in the drink up on our horse when he was still a youngster. She rattled the ice around and took off, dumped her on her head and put her in the hospital. Crinkly plastic sandwich baggy type things will get the same reaction. If she wants to do anything but ride, she needs to get off and do it on the ground.
You need a good trainer who specializes in desensitizing the young horse, not just training them to jump or lunge or canter, but will do all the things that just got your daughter dumped, until he's not spooky anymore. This all sounds way above your pay grade (and your daughter's) and is a good way for her to get hurt by his not so "unexpected" behaviors.
I've been given to different types of sage advice, and the older I get the more true I find them. #1 Horses are only afraid of 2 things. Things that move and things that don't. #2 Horses only have 2 thoughts in their heads when they wake up in the morning. What shall I eat and how shall I kill/injure myself today.