You have three issues here – the first is social, the other two political and of course the first impacts on the others. There’s no easy way around the first issue and I think you have to be firm (but not too aggressively so) and tell your neighbour that that you appreciate her concerns but she is intruding on your life and making you unhappy.
There are too many interventions and what she says is not only factually wrong but distressing to you because she is accusing you of a serious crime – mistreatment of an animal. You don’t want to have any more discussions about the cat. End of story. If she feels you are mistreating it, then she can call the RSPCA and let them decide.
Meanwhile, her constant contact with you is not what you expected when you moved in and you would like her to back off a little. You want to be a good neighbour but, popping her head round the balcony is an intrusion on your privacy and you want it to stop immediately. You want to be a friendly neighbour but this is too much.
If you have trouble saying that to her face, go and get a nice card with a pussycat on the front and say it in writing then slip it under her door. Practice writing out what you are going to put there before you actually put pen to paper so you can get the tone exactly right.
If it continuesafter that, then you need to explain in no uncertain terms that this is your home and you are entitled (if you are) under the by-laws to “the peaceful enjoyment of your lot”. Being constantly harrassed by her is a breach of the by-laws and you will take it further if you have to.
The political question is trickier. Often this type of particularly obsessive EC chair imagines the building can’t survive without them and will periodically announce that they are going to resign at the next AGM, in the hope that everyone will fall to their knees and beg them not to do it. Old hands will do so because they know that not to can lead to all sorts of shenanigans and so the cycle continues. The chair feels needed and feels that all their behaviour has been endorsed by the committee members.
It may be worth talking to some of the other committee members before the next AGM to find out iif there’s any feeling that they would like her to quit. Be subtle and bear in mind that by both telling her to back off AND finding out if there’s any chance of a ‘palace coup’ could look like an all-out assault on her. Maybe one thing at a time is enough.
As for being on the committee is concerned, I would hang in there, make sure you are familiar with your building’s by-laws and be ready to make a move. If she ever says she doesn’t want to be chair any more – at a meeting at which an new chair can be elected – jump in. The most common reason for a control-freak chair being re-elected is that nobody wants to step up and take their place. Maybe that person is you – but you can’t do that if you aren’t on the committee.
It’s probably worth getting to know the other EC members anyway – maybe one of them is close enough to your neighbour to intercede and prevent this turning into a feud.
Best of luck with this. The worst thing that could happen is that she just won’t speak to you any more … and would that be such a bad thing? By the way, don’t worry about permission being rescinded – even if she can bully the EC into agreeing to that, they’d have to jump through an impossible set of hoops to make it happen. Don’t forget, you are probably not the first person to discover your neighbour has a few ‘roos loose in the top paddock.