I agree, to some extent, with marialuisa. Don't make big changes to tack while pony's settling in, but I'd have her out of the gag immediately. Take off the flash in a safe environment when you're ready and see whether she opens her mouth. If she does get her teeth checked. Only use a flash if the pony is genuinely evading the bit.
When I got my mare the person I bought her from was riding in a flash because she was opening her mouth, yet the woman knew that her teeth were neglected because she negotiated the price when she bought her because of it, then didn't get them done. Teeth fixed, no problems (sorry, bit of a rant there)
What you're describing about her head carriage is probably the result of her being ridden in a gag which exerts poll pressure and pulls the nose in rather than creating a correct self carriage so she needs to learn to lift and work through her back, engaging the hind legs. The way to control a forward going pony is through seat and legs, not hands so make sure dd has a good deep seat and encourage a long leg. This will aso help to develop the ponys way of going.
Your instincts are good, transitions will help and your dd is dead right about neck stretches, pony needs to go long and low and reach into the contact as the first stage to learning self carriage.
Re being forward going, that's a good thing as long as dd can remain in control, which she sounds as though she can, but I know what you mean about it being a bit scary as her mum. Don't worry, as you get to know and trust the pony you'll relax.
Keep us posted, she sounds lovely, and your dd will love her. GD's connemara/tb was a bit like this when we got her, and she's much softer now. Schooling, lessons, more schooling, more lessons and lots of hacking if it ever stops raining. And some jumping for fun, but only after a good schooling warm up. (I think you have good instincts and know more than you're giving yourself credit for)