Shared chimney stack removal

[wellington35]

Hi - newbie post!
I've done loads of online research and had one quote so far but the more I find out, the more questions I come up with. Hoping someone can help me be more knowledgeable before I get another company round to quote/advise.

Mid-terrace 1890s 3 bed. Shared stack needs repointing as causing leak to neighbour. Neighbour has no chimney breast there so assume it's got support between 1st floor ceiling and loft (hope!). On my side I have no chimney breast on ground floor and no idea what's holding everything up. I want to remove the stack when doing other roof repairs. In a few months I'm wanting to remove that chimney breast in the first floor bedroom so I can relocate the bathroom. Talking with neighbours over cost sharing and party wall agreement. Questions are:

1. Should I get a roofing company to quote for the whole stack removal to below roofline?
   
2. Would support of any kind be required for that?
3. Is it better to get the entire structure removed in my house down to the loft floor (or would it need to be both properties) and then I can take out the chimney breast in the bedroom/bathroom later?
4. What's the building control situation with these options? Any idea if it's one piece of work or would all be part of the one job??

Any info, advice or other questions I should ask, would be much appreciated. Sorry - for long post!