Both dc play piano and one other.
DD1 says sax is cool for a boy. These can be in jazz bands and some orchestras.
Clarinet def in both but many people play this so may be harder to get into ensembles with this unless you are a high grade in some schools.
Bassoon is not in jazz bands and costly instrument. Some areas I have heard have cheaper lessons or funding (one of these - can't recall where I heard it ) for less common instruments including bassooon, oboe and cor anglais.
Depends on what your tolerance of noise is like. Certainly even a good grade (eg grade 5) I think can sound awful on violin (purely my personal opinion but I've heard plenty). Cello more forgiving but transport can be an issue - cert if bus involved.
House size, detached or otherwise may be an issue. Certainly a trumpet may not be popular with neighbours if you are in a semi or terrace. Ditto drum kit. I am on quite a wide road and when first moved in looked for radio left on or thought I was hallucinating. Turned out to be son on opposite side of road.
There is a book called 'The right instrument for your child '. Looks at personality, dentition etc of your child and tells you about each instrument. Mine was lent to music teacher and not returned.
Book had interesting titbits e.g bassoonists are classically the jokers of the orchestra, the french horn is for the intellectual of the brass section and the trumpet is for the extrovert as everyone can hear if they make a mistake. Don't know how true certainly the latter is as the 2 good trumpeters I know are unassuming. Maybe they come out with their instruments.
You are right in that there is a lot to consider.