They are two steps here, first to get the on-site interview and then to pass the interview. You talked about problems in both of them.
First, how to get the interview. For me, the trick is to get a reference and apply to the right position. You have specific background in some languages and environment. It is hard for the companies to decide where to put you. If possible, start with networking with people in your target companies. Get to them to know you first, they will help you on finding the right position to apply to.
About the Twitter manager who didn't reply. Try again, with other methods. Find a direct reference instead of simply email him. Or you can hang out with some other Twitter engineers first and they may help you out.
Second, for the interview part, there is little description here so I am not sure if you did anything wrong. Generally, solve the problem by working and discussing with the interviewer; showing your passion; and relate yourself to the company. It is hard for the interviewer to see the value of your project and weigh them during the interview, solving the problems they give is more important.
Also, there might be a bias since you have the major bank background. I cannot elaborate or back that up though.