There's two, or even three separate issues which need to be addressed here.
#1) Interpersonal relationships
It sounds as if you guys are living in rather tight conditions. It's typically disastrous to pack 2 immature teenagers into such a confined space, with no room to avoid each other, and get over the various issues and pressures they are each feeling.
During this time of their lives kids are typically going through quite a bit of stress due to school, hormones, social pressure, etc. The fact that they are constantly stuck together is probably a constant aggravation, like an itch neither of them can scratch.
It takes a lot even for a mature, calm person to live with an individual they dislike. What you need to do is sit them both down, maybe separately at first, and explain that they need to start getting along. Get each of their version of events, and try to reach a compromise. Explain that you are all far away from home, and need to work together in order to succeed.
Ask your sister to count to 10 in her mind, and breathe deeply before she simply explodes and start yelling at him. Explain to her that there's no point wasting her energy getting, and being angry all the time. Teach her to learn a little bit of detachment from the situation - not everything your cousin does is meant as a personal insult to her.
As far as your cousin is concerned, get a short list of behaviors which really upset your sister, and ask him to correct those issues. Ask him to make an honest effort to improve those 3 or 4 bad habits, so that there will be less friction between them. It's important to also explain to your sister to not blow up if he makes some mistakes as he is trying to change his behavior, but rather work with him in trying to overcome those habits.
I know this is all much more difficult to accomplish than it was for me to say it, but I think you can get some real results if you can get them both to cooperate.
#2) School work
Your cousin's poor school work doesn't come as a great surprise. He is a young kid in a high stress situation, in an environment he may not feel comfortable in, and lacks parental supervision. Left to his own devices he is certain to simply slip further down the ladder, rung by rung.
He requires a parental figure to step in and correct his behaviors, starting now. I know you're a student yourself, and have a lot of pressure to deal with, but you're going to have to step up and give this kid a little more structure and guidance.
Simply handing over a textbook and internet connection is not going to do it. It is, in fact, going to lead to more procrastination, as you've already seen.
Since his English leaves a lot to be desires, start by speaking only English with him. Your sister and you should both only be speaking to him in English, and asking him to repeat any sentences that he says in your native tongue. Simply don't answer him unless he speaks English. Correct any poor grammar as he says it, otherwise he will retain his poor vocabulary, and not improve. i speak from personal experience: I had a Korean friend who, over 10 years of living in Canada, barely spoke decent English. I corrected him relentlessly, to the point where he wouldn't even want to speak to me anymore, but later thanked me when his English improved by leaps and bounds.
Now, as far as his homework is concerned, a lot of his poor performance is probably related to his difficulty in dealing with the English language. As that starts to improve, so, most likely, will his marks, because he will be able to understand what's going on in class, and read his material with a higher level of understanding. However, there's another aspect to it all, which is his own personal sloppiness, and irresponsibility.
Encourage him to write down daily and weekly lists of homework assignments due, homework, or upcoming tests. Ask him to cross off all the items for that day before he watches cartoons. Encourage him to do better by congratulating him when he crosses an item off the list, and maybe reward him when he gets whole lists done (buy him an ice cream cone, take them out for pizza, etc.)
When doing this, explain to your sister that your cousin lacks the discipline and motivation which she has, and that you encouraging him doesn't mean that you're favoring him over her. She should try and help him as well, but if they dislike each other that much it might just be better for them to leave each other alone at first.
Build that sense of accomplishment in him every time he does something for school. Tear him away from his cartoons not by threat, but by challenging him to do better. To rise above his current apathy. Remember, he will not improve unless you actively get him involved in the process.
