I'm having problems with a neighbor for the first time in my decade living in New York. She came upstairs to pointedly complain about my dryer being on at 3 pm on a Saturday. My response was to tell her we were willing to accommodate her as much as possible, but that my laundry also needed to dry. We heard from her again on a different weekend when she could hear a tray table moving while my husband was cutting food during dinner. I was told she was nicer about it this time, but I also know she's complained about noise to every other apartment on the floor. We are insanely quiet neighbors because we are aware that we aren't the only people in the building; we keep our tv down, we don't shout, we don't stomp, and I'm allergic to making noise after 10 pm. We also live on a busy street in Queens, so the expectations of absolute silence seem off.
I could deal with a stroppy neighbor if inconvenient and unreasonable requests for quiet were the only issue, but then there's the roof deck. We get access to the small common roof deck that also houses the building's trash cans. The day this woman moved in, she started treating the deck like private storage, and most recently I went upstairs to find the ugliest, least suitable patio set imaginable just sitting there, blocking the route to the trash and taking up the last available inch of space. There is now no room for me to use the roof deck the way I want to (or for anyone else to for that matter), and the deck feature was part of the reason we decided to pay the high rent for our tiny apartment. There has also been no indication of whether or not it is intended for group use. I know she's responsible because the box for it was just thrown on top of the garbage cans and had this neighbor's name on it.
Now there are a few ways to approach this. If you prefer the 'catch more flies with honey' approach, come to her with (what really does seem like, from your letter) an honest and true interest in making the roof deck space as usable and nice as possible for all involved. Confirm that the patio furniture is for sharing and maybe offer up other ideas you've had. But above all, bring up the need to access the garbage cans and your desire to keep the trash area clean. Hopefully the common New York language of not wanting rats will reach her.
If you want to keep things strictly business, refer to your lease. Oftentimes there are notes about what can and can't go on outside spaces and rules for keeping the trash area clean. You can politely remind her that you're all beholden to the same rules and you'd like to make sure the space is up to code. If things get out of hand, you can bring in a super or landlord, but try and keep things as direct as possible.
As for the, well, personality clashes, you can only do so much. You are a good neighbor and you know that, but at some point you have to take a step back and consider how you engage with this woman. Are you upholding the unwritten laws of basic neighbor decency? Are you polite? Great. Set as clear boundaries with her in regards to noise and space as you can. AND THEN, live your life in a way that suits you, you wild afternoon-laundry-do-er you, and give this woman space to grow up and get over her own stuff. And if she can't, let's hope she considers a move to the country.