Arelith

Personally I've always made an attempt but it's never really gone anywhere. Either no one is home, people don't answer the door or aren't seen outside of their immediate home, or some other similar reasoning. The only time I've been able to have neighbourly shenanigans was with people I knew out of character and happened to bump into. I much like the idea of it regardless of never being quite successful with it.

I do!
My current character knows/is friends with all of her neighbors in the zone she lives in aside from one person. Most of my characters have been that way with the exception of my bard, which was because I never saw the people who owned the quarters next to hers.

I stare at the entirety of the Abyss. I stare at you. I stare out into the abyss again.

I am so lonely.

This one varies a lot for me. When my main was a less known character I generally tried a little more to know my neighbors in the places I occupied. Their current home though was purposefully selected because it's out of the way and just because of how tied into certain regions my main is I try to avoid actively seeking my neighbors to give them "room to breath". That's not to say I'll actively refuse to engage with my neighbors if they write me notes but I won't for example check for new arrivals like I might have in the past because I want to give me neighbors a little room to be disengaged from me while they themselves get established.

Depends on location of the quarter.
But mostly yes my current quarter only has two others nearby and I've never seen either character in my time zone.

Eight (8) quarters in a relatively small area... Each one is occupied and every single one of the owners interact with each other.

I certainly try to! Granted, I'm presently under an obligation due to my character owning a Guildhouse, but even beyond that, I'd try to interact with neighbours as my characters see them. Even if they never cultivate a personal relationship, there's definitely that neighbourly "Hello!" and being able to say "Shankswell Guttingham? He's my neighbour! I don't know him very well but he's always so chipper and polite when I see him!" and the like.

I answered "I don't"... But it's not fully true. I got my current quarter thanks to a IG/IC friend and I used to interact with him. But most of the time we interacted outside of the housing area where we stay.
My char has been approached recently by other characters living in the same area, telling him that he should spend some more time around the village and get to know the other inhabitants. He (and I) would happily like to do that. Unfortunately, most of the time when my character gets out of home there's no one around, so he heads to the cities, and generally comes back when it's time to sleep.

Sil'avesi sees it as a duty as someone who cares fo Guldorand to somewhat know her neighbours and the overall city. To keep them feeling welcomed she seeks to help and talk to those she knows in Guldorand. Part of her overall long term view of securing the peace of Arelith is to secure the social comfort of her neighbours feeling welcome.

It depends on what character I'm playing. All of my characters wanted to know who their neighbors are, but some of them would invite their neighbors over for dinner regularly and others would just want to keep an eye on them in case they were shifty.

i like to rob them

I was in my quarter, when I heard the door close nearby and then there was sawing sounds.

So I went to knock on the door, and when they opened up, I told them if they heard someone sawing, and how annoying that was - They were terribly sorry and then I told them to keep it low because I wanted to sleep (14 pm).. they said sorry, I said it was ok, they brought a gift the next day. It was good.