Every response to your initial question seems to revolve around the one subjective part of the introduction of the wiki article on neutrality:
"Above petty conflict."
I went ahead and removed that part, as the wiki is supposed to be a resource and should be as objective as possible. That statement has, at least, two things wrong with it:
1. It presumes, as it seems a lot of people do, that to be neutral is to believe that you are above conflict. This is a flawed notion for many reasons, but the one I'd like to focus on is because neutrality doesn't protect you from conflict. It offers you (some) protection from violence. Conflict and violence are not synonymous. Human and supernatural society could not progress without conflict. It could progress without violence (some people would say more slowly, given the economic growth and technological advancements afforded to human society by some wars).
While a neutral person may believe that they are above violence (to use the correct word), that isn't the rule for people who take neutrality. And people who believe that this is the only reason to take neutrality, let their bias against neutrality leak through.
2. The word petty is
extremely subjective and again, adds a negative connotation both to the person taking neutrality, and to all of the violence in this city. You seem to use the words "conflict" and "violence" interchangeably, so the next part of my statement is pure speculation.
You've stated "We all know (or should know) that conflict is built into the foundations of this city, and that Vampires can be cruel" and, again, I'm assuming by conflict you mean violence, seeing as conflict is built into the foundation of every society on some level, whereas violence being built into the foundations of a city is probably only true for RBC (if you believe your hypothesis) and Sparta.
If some amount of that violence is petty (some people would argue, based on their personal definitions of petty, that there hasn't been a "glamorous" or honorable" bit of violence in the city for years) wouldn't that make some of the foundation of this city petty? If you believe that a majority of the violence that has happened in the past few years is petty, than our city isn't built on a foundation of violence, it's built on a foundation of pettiness.
I think most of us would wholeheartedly disagree with that hypothesis. I personally also disagree that violence is built into the foundations of the city. The city was founded in July 2002 and the first act of aggression against non-humans wasn't until December of that year, and there was even a vampire who tried to stop the violence (this is how the Nightwatch was born). The very first act of organized movement on the city grid was aimed to
stop aggression.*
I would argue that the foundation of this city was that of a haven to beings like us who were lost, displaced, refused, or for some reason didn't feel comfortable or safe in the rest of society, and we found refuge in this city. The violence only came when the violent people came.
All of that aside, my opinion on each of your questions in turn:
Do people stop and ask themselves, "do I want it to happen to me, or not?"
I've thought about this many times. I can't speak for anyone else.
And is there something wrong with that?
I don't see why there would be anything wrong with anyone thinking anything.
Is there a stigma against people taking the Neutral option? Do people even know it exists?
I know it exists. And, based on two of your responses, I would say there is a stigma against it. It appears that the stigma against it is someone who takes neutrality sits on a pedestal believing they're better than everyone who chooses to engage in violence. I don't personally care either way if someone takes neutrality. If someone took neutrality and took to verbally harassing people and thus exempted themselves from the traditional sense of punishment for such things, it might be a different conversation.
Is the fact that we are all still fair game (more so once we have Full Power) being considered and taught and the appropriate choices suggested and encouraged? Should they be?
I don't think it matters what's being taught. There are ways to lessen your chances of being hit or zeroed, but there is no way to guarantee it (assuming the wiki article on neutrality is still up to date and you still take damage from weapons, just significantly less). You can look at someone for a bit too long on a day that their significant other(s) left them and you're going to get hit. There is no "proper etiquette" as any etiquette will get you hit, depending on the day and any given person's mood.
Is it being considered that everyone is fair game? Probably. I'm sure that idea is what keeps the very thin line of order between families and clans. Do people think much of it? No, that's why the line of order is very thin. Needless, but for some reason necessary, I'll let you sort out the paradox, to say, this is all my opinion.
*http://en.ravenblackcity.com/wiki/2002