Kinda biased because where it's where I live but in general it's a nice place to live. You're going to want to get a decent raincoat because it rains a LOT.
You said Newtown by the hospital. Is this really Newtown or is it actually Raffles i.e. the area around Shady grove Road A lot of people will say Newtown instead because Raffles really isn't very nice. It's a large council estate and definitely has it's problems. I would avoid living there (if you PM me your prospective new address I can let you know if you want)
Yes there's a gymnastics club. There is also a thriving trampoline club in Sandsfield Park. There is also a pretty decent swimming pool in the Sands centre.
In terms of schools they aren't that bad. Newtown would be either Caldew Lea or Newtown depending on where you are. However with the exception of one or two schools you can pretty much pick where you want to be.
Public transport is actually pretty decent in Carlisle but it does finish early (1030 on a Saturday and around 1930 on a Sunday). Traffic is pretty decent i.e. not bad unless the council are digging up the roads again.
Bad bits: Carlisle (and indeed the rest of the county) is absolutely dreadful for SEND education. There are very, very few special schools, hugely oversubscribed and only the most complex children get a place. Everyone else gets shoved in mainstream education which really isn't working for a lot of children. Also, while there is a children's ward at the hospital they don't take very sick children. Anything remotely 'interesting' will get you sent to the RVI in Newcastle. In the winter the road (A69) genuinely can close because of snow in certain places. I've known it happen that acutely unwell children get a helicopter ride. It's getting better but anything hospital paediatrics is usually dealt with at the RVI.
Also, town pretty much stops dead at 1730. Everything closes then. There really isn't much if at all anything of a nightlife scene. Even the takeaways tend to be closed by 2230/2300.
Good bits:
It's a very friendly place to live and people saying 'good morning' to you as you walk down the street is absolutely a thing. Also there's a very neighbourly feel. During covid, a grassroots organisation did a lot of good work in the city. During the 2005 and 2015 floods again the city clubbed together.