I have a drunken mother, a father who can't control her, a brother and sister who are afraid of me because of my temper,

You can't do much about your mother unless you have amazing persuasive skills to help her stop or send her to alcoholics anonymous. You should sort your temper out if you want a better relationship with your siblings.

Sure, I have things, like a laptop and an Xbox 360 and all that, but why can't I have what I want.

That sounds really spoilt tbh. Be thankful you have food and the basic things to keep you alive. Some people don't even have family.

I was not chosen for an NJROTC related event that even my Sargent Major said I was qualified for

That's unfortunate but that's just life; unfair. If you believe in fate then you should consider this to be something which sets you back on purpose to make way for something better. Things go wrong but you should remember the things which go right. Everyone has setbacks but you have to just shrug your shoulders and move on and try again.

I am too much of a weakling to ask out the girl of my dreams,

You're not a weakling, you're just shy which everyone is. You just have to go for it and remind yourself that you have nothing to lose but a lot to win! She's not going to laugh in your face (unless she's immature, in that case, she's not worth your time) because if she's nice enough, she'd give you a chance. If you're scared of doing it in person, do it through FB, it makes it a bit easier.

It just feels as if the world is against me.

I know this feeling, I think everyone does but the world isn't going against you. I've been feeling it recently. I dislocated my knee at my first every kickboxing grading, so I can't walk for a while now, I have an awful cold, I finally got a wheelchair so I could go out to enjoy the weather but it rained when we were out, then I wake up today feeling even worse and not up for going out but the weather is beautiful today. It's not the world against me, it's just coincidence and unfortunate, it happens to everyone. You just have to stay positive for the future and hope for the best. Remember, you're not the only one going through rough times and there's always someone going through a whole lot worse.

Sell your Xbox and TV, then use the time you would have been on those in order to accomplish things you want to accomplish.

yeah you need to get rid of distractions and focus on your life. forget about all the worry and stuff
as for your mother you cant help her bad habits only she can help herself. your dad is probably trying over and over with no luck. you cant help your anger issues although you could try and work on them. try and fix things before its too late man

For anger issues, may I suggest something? Btw, my drunken father left when I was one thankfully and I also used to have these issues of anger. Anyway, what type of music do you listen to? I used to have severe anger bursts and depression when I was in middle school and first year of high school, attempted suicide, then I attended therapist/counseling lessons. One recommendation they had was to listen to uplifting music and from that day on, that is the only thing I have listened to and I have become a completely different person. Everyday for a couple of hours I put my headphones on/speakers and I just finish homework, work, or relax/nap listening to the music.

Try these Youtube/Soundcloud Channels out and enjoy! (copy and paste into browser)

youtube.com/watch?v=h60C0Ubrp5o&list=UUSXm6c-n6lsjtyjvdD0bFVw
youtube.com/watch?v=Fsy7R68wnLM&list=UU5nc_ZtjKW1htCVZVRxlQAQ
soundcloud.com/silkmusic
soundcloud.com/anjunabeats

Start doing some sport for all negative energy (temper issues)

When you feel that you want to hit someone, start doing push ups. Do them until you can't even move. Then just lay in bed (maybe shower before laying haha) And you see, all negative energy is gone.

I also have temper issues, had intention to hit things and just burst into anger.