No, you don't need the martial arts specialization to use martial arts qualities. The specialization is just a +2 when using martial arts in unarmed combat. You technically don't even need the Unarmed Combat skill to take the qualities, which makes sense for at least some of them, like Firefight, which doesn't necessarily use the skill at all.
From a build strategy, your logic and charisma are eating up your points and giving nothing back. 2 charisma +1 Influence is 20 points that actually hurts you more than it helps you. With only 3 dice for social, all you've done is set yourself up for critical glitches. Those 20 points are likely to get you chased by a torch-carrying mob, rather than actually see you through social situations. I know it's realistic that a person would go through life learning at least a little bit of social skill, but the dice-based system punishes that kind of realism harshly. It's better to have a 1 charisma and no social skills, and to not even try to roll, than to have 3 dice that blow up in your face. If you want to have some kind of decent social pool, what I would do is buy your social pool up to 4, and then get rating 4 empathy software from Arsenal, so you'll have 8 dice, which is enough not to glitch every time at least. Buy up charisma first, because attributes are expensive after chargen, and skills are much easier to acquire. You should also consider taking just 1-2 social skills instead of Influence, chances are you don't need, say, leadership or con, you can get by with etiquette and negotiation alone, since no matter what you're not going to have a majorly impressive social pool.
The Stealth group is also a big waste. You do realize it includes shadowing, disguise, and palming right? Those are hardly useful for a sniper. Snipers don't shadow, they hide and wait. Disguise doesn't matter because you're hiding. And palming is neither here nor there. I'm not saying the skills aren't useful, but would they be part of sniper school? And more importantly, do you really need them? If you want a combat character, you need to tweak your skills so that you kick ass. Too much versatility is poisonous to ass kicking. If you want to be at all good at your profession, you need infiltration to be a 4(6) at least, a 3 won't cut it. The rest of the stealth skills will not help you enough to be worth the expense.
Your next biggest waste is Logic. It is a useless skill in this game, pretty much, unless you're a tech-wiz type. The only logic-related stuff you have has very limited utility, and you're wasting 30 points on the stat.
A Dodge of 1 is really dangerous. You're just asking to get shot to death. I'd raise it, especially at the expense of logic, charisma, and linked skills. You can't out-think or out-social a bullet, but you can dodge one. Leave the thinking and the social to the less combat-oriented characters, if you try to do too much, you will be a joke, and people will wonder why they're giving you an equal cut of the pay.
Outdoors and Athletics groups are iffy. Outdoors especially, since Shadowrun is in an urban setting. And Athletics includes questionable skills like running and swimming. I've never EVER seen anyone swim in Shadowrun (ok, I have once, but the character swam to shore just fine without knowing the skill) and running doesn't matter. In fact, all that swimming and running do is make you swim and run faster. You can swim and run without knowing the skills, you just don't get a test to go faster, which you really don't need most of the time. However, you can probably afford those groups at level 1 if you move other things around, and it would make sense for a character with some military training or w/e, and it might not end up being totally worthless.
As for your negative qualities, they seem problematic. I'm all for power gaming, but Lost Loved One is the biggest cop-out of all time. It's free points. It makes sense in tabletop, where the GM can design runs based on your lost loved one. Maybe he drops a hint about them, and you have to go chasing after them on a quest with no reward except maybe finding your lost love. But in a MUSH, the runs are just pickup games, the GM usually doesn't design them with your character in mind, it's just plug-and-play. There is virtually no chance that flaw will bite you in the ass, which means IMO you shouldn't take it. Feel free to have a lost loved one in your background, but don't take points for it, it's just unsportsmanlike.
Also... your character hates gays, but his lost loved one was his spotter? I dunno about the 2070's, but today, the spotters are all dudes... Maybe there's a little hypocrisy in this guy's story?

As for the prejudice, I don't share Green's antipathy towards the prejudice. First of all, it's just a level 1 prejudice, so he's more of a grumbly-frowny prejudiced than someone with an actual full on hatred. And IMO it's perfectly fine to create flawed characters who have something nasty and reprehensible about them. That's why flaws are worth points, they get you into trouble, and being prejudiced against gays in Shadowrun's gay-friendly culture is likely to land you in hot water at some point. Especially if you're in loved with a lost dude but can't admit it

The other qualities seem ok probably. A sniper could learn krav maga cuz he feels like it, or he could be Israeli, or another military might have adopted the martial art, who knows? The compulsion could be a big mistake, depending on what it is. It has to be something that sucks, because it's a level 3 compulsion. If it's something like compulsively twiddling your thumbs or something, it just a cop-out to steal extra points. But you have to be careful, because a compulsion that's harmful enough to be worth points is often not conducive to a successful, live character.
Green's comments on your cyberware are all correct, of course. I will say this: the biggest favor you can do yourself is spend 5 points on Restrcited Gear and buy Muscle Toner at level 4. Having Agility at 8 or 9 makes the hitting part of combat a breeze, and once you have that rolled up, you're doing ok.