I came to London for work in 2013. (The standard phrase is "for work".) Since I came to New York, I have been working as a computer programmer. I will be waiting here until you come back. ("I will / I'll wait here until you come back" is the more natural choice of tenses. The use of the continuous makes it sound as if you are trying to stress the momentary nature of the waiting throughout, as if you will be anticipating the return of the person to whom you are speaking very much.) After you come back home, I will cook something for dinner. ("After" sounds a bit forced, as if the person you are talking to you just said "cook something for me before I get home" and you want to stress that you will only do that AFTERwards. The neutral sentence would be, "When you get home, I will cook something for dinner".)

I came to London in 2015 I have been here since 2015. I have lived here since 2015. I came to London TO WORK in 2013. (You must use the infinitive here, because at the time you came to London, you had not completed any work in London yet.) Since I came to New York, I HAVE BEEN working as a computer programmer. (Use the present perfect because the action has continued from the past into the present.) I will be waiting here until you come back. After I came to London, I started working as a programmer. After you come back HOME, I will cook something for dinner. (We almost never say "to home" in English.) After I FINISH my homework, I will help you. (You have yet to complete the action, so you should not use the gerund form "finishing.") I hope I've helped. Please let me know if my explanations are confusing.