Its not a permanent structure though, no footings etc. They are just using their garden, legally 11 pm is accepted as shut up time, and they are by your op so not a lot you can do. You could talk to them yourself about how it is effecting your DD but you probably know already how that will be recieved by them and whether i will make any difference. They aren't breaking any laws so its only good will left to go on.
Completely within your rights to tell them to shift the bit sticking over to your boundry though.
There are no rules about temporary structures. However, if it is physically invading your space and your neighbours are being antisocial, then you've every right to complain. PS A match might solve your problem
The normal rules are that a permanent structure has footings, and that it therefore needs planning permission. A lot of pub 'smoking pagodas' fit into this category.
Your neighbour's [flammable] gazebo isn't a permanent structure, but you can probably challenge them over the noise nuisance, though, tbh, either by speaking to them yourself or complaining to your council's noise team (Environmental Health).