The short answer is yes, but you should not take legal advice from people on the internet.
The long answer is, you should really talk to a lawyer if you have questions about self-defense. Jails are full of people who thought they were acting in self-defense, but crossed the thin line at some point because they didn't know what exactly it means, like you. Consider the following a rough guideline on principles that are usually valid in most countries, but check with a lawyer before doing anything stupid.
First of all, you should understand that when you claim self-defense, the burden of proof is now yours. You're no longer innocent until proven guilty. You're admitting that you attacked someone, and now it's up to you to prove that you were justified in doing it because you were just defending yourself.
There are five principles that determine what is justified in self-defense. Innocence, imminence, proportionality, avoidance and reasonableness. In very simple terms, what they mean is:
Innocence - You cannot initiate a conflict and later claim self-defense. This means that you'll have to prove in court that you didn't started the fight.
Imminence - The danger you are defending yourself from is something happening right NOW, at the moment! You cannot claim self-defense if you are reacting to an hypothetic danger that might happen later, like a verbal threat, or a danger that is already over, like hitting someone after they are incapacitated on the ground. You'll have to prove that you were in immediate danger.
Proportionality - The degree of force must be proportional to the threat. You cannot break the arm of someone who touched you with a finger, or shoot someone who punched you. You'll have to prove that your reaction was proportional to the threat and you weren't overreacting.
Avoidance - There was no other way for you to protect yourself from danger other than reacting with violence. This means that if you have a non-violent option that doesn't increase the danger, you should try it first. You'll have to prove that there was no way to avoid the fight.
Reasonableness - This is a subjective principle that applies to all the others. It means your perception and reaction regarding all the others must be compatible with that of a reasonable person under similar circumstances.
Now, to your question:
Am I allowed to use Krav Maga for any case of self-defense?
You are allowed to use anything for any case of self-defense, as long as your claim of self-defense follows these principles, which may vary in many places. For instance, many states in the US have the so-called "Stand-Your-Ground" laws, where you don't have to prove that you were cornered and had no way to retreat. Also, reasonableness is very subjective, and what sounds reasonable for a judge or jury in one place might not be reasonable somewhere else.
Even a threatening or grave suspicion.
It depends on what you mean "use Krav Maga". If you mean engaging in a fight with someone, then absolutely no! Following the principle of imminence, reacting violently to a threat or suspicion is not self-defense. There are a lot of people in jail whining that they were just defending themselves when they attacked someone who they thought was threatening or suspicious.
I live in Brooklyn, NY, but it would be nice to know how this works in other places too.
The only sane answer here is: hire a NY lawyer for one hour of consulting and ask how exactly NY laws work in self-defense claims. If you are going somewhere else and you believe there's a chance you might find yourself in trouble, check the local laws before you leave, specially if you're going to another country. Trust me, you don't want to be in a foreign police precinct after beating a native for what you thought was self-defense.
If you don't get professional legal advice and you ever need to use Krav Maga or anything else in self-defense and you end up doing something serious, like gravely injuring or killing someone, call the police, rescue services and a lawyer immediately, and don't say a word to anyone until you do, not even to claim that you were acting in self-defense. To the police that's a confession of assault or murder, and they usually don't care about your self-defense claims other than to put it on record. It's up to the justice system to decide if you'll be prosecuted. If you don't know any lawyer -- and you should, if you are training self-defense techniques and planning to use them -- simply ask the police for a lawyer. I can't emphasize how important this is. A lawyer will know if you have a solid self-defense case or if you have better chances with some other strategy. You don't want to end up in jail for defending yourself, but many people do.
and as a plus, how the heck can you walk in a ready stance down the street without the whole freaken place going 'ooooh, aaahhh'.
Seriously? You don't. That's nonsense. That's not something a normal person sitting on the jury hearing your case will find reasonable. The opposing lawyer will have a great time convincing the jury or the judge that you are nothing but a freak who walks the streets trying to pick up a fight for no reason and claims self-defense later.
However, there are many natural non-threatening stances that are quick to shift into a defensive or aggressive stance that you can use in a potentially threatening situation. For instance, holding your chin with one hand while supporting the elbow with the opposing hand like this is a natural, non-threatening position, that allows you to defend your face and your genital area easily, which are common targets for someone starting a fight. Read Kelly McCann, Combatives for Street Survival, he gives some advice on that.
But please, don't walk around the streets in a ready stance like you're looking for a fight, unless you are trying to get your ass kicked.
You say you are training one-on-one with a partner, so it doesn't sound like you have an instructor. Learning the mental and legal aspects of self-defense is as important as the physical and technical aspects. If you have a instructor and he never taught you that or never hired a lawyer to give a lecture on that and answer questions, I would recommend asking him to do it.