Others will know the detail of this better than I do, but my lay understanding of the law is that child is deemed responsible for criminal responsibility from the age of 10yo. There is a point at which a child is deemed to be capable of responsibility for their actions. Of course, children develop at different rates around this age, so some mitigation is taken into account. It’s not always straight forward.
So @MumOfFury understandably you are angry, upset & alarmed by what your child has done let alone the actual cost of making good.
Would the bathroom door usually be locked by a child at home?
A locked bathroom door is a non-starter in our house, so privacy when the door is closed is strictly respected. Is a lock on the door something new so they used it without thought?
I am a bit confused about which door was kicked in, by any measures this was unacceptable. You mentioned that you told him what to do, e.g. stop, through the locked door which meant that you could not be there to intervene, & he carried on? Then to put something, let alone a fabric item, on top of an open fire is plain scary.
Is your child this reckless at home? Do they usually need much supervision?
I think10yo is old enough to learn that there are financial consequences to actions that damage other people’s property. This is something that needs to be aired & discussed. The savings you had in hand for everyone’s Christmas may now be gone, so fewer presents/less choice etc all round.
Denial of screen time for a fixed period has scope. Can you cap their data use so that there is a ‘nominal’ repayment. Maybe a limit of new books, comics, treats etc for a fixed period?
The discussion is why did they carry in doing something - not a little bit naughty or indulgent, but active damage -when told by a parent not to do so, & why do something dangerous - heat, fire, flammable - when they know not to do so at home?
The outcome may be as parent that you may need to revise how much risk & responsibility at this stage that your child can understand or manage, e.g. access to/not breaking stuff, walking home from school, asking for help to resolve things that are a bit beyond them at the moment.
Lastly, no matter how mad or upset you are, please don’t send your child to your mum’s at this time. For you it may seem like a break from your difficult feelings, but actually is it a form of exile/banishment/rejection for your child & this may come back harder & harsher on you both later on.
First work this matter through together at home, let the dust settle & then maybe a few days with your mum so that you have a bit of space to yourself. A decision to be considered with a cool head & a calm heart.