I don't mean this as criticism of OP's work, but if you want to influence your representatives you're better off calling their office or sending snail mail (postmarks show that you're a real constituent).
You're right, and this is of course meant to supplement those efforts, but as a political activist, I've noticed more and more delegates and senators, at both the state and federal level, relying more and more on Twitter. If there's any traction at all, the next steps are to add the House members, and also allow for emailing at the same time.
What I've found is that on hot-button issues, like gun control, CISPA, gay marriage, etc., the majority of your emails are ignored, and are just parsed for whether or not it is 'for' or 'against' a given bill. Twitter allows for an easier aggregation of that sort of content than email does, and forces us citizens to keep the message short and on point.
Namely, this is meant to address the audience that doesn't know who their elected officials are, and give them an easy way to voice that opinion.
>sending snail mail (postmarks show that you're a real constituent).
The people I know who have worked for US senators strongly disagree with writing letters. For one thing, there is generally a two week delay, so if you're writing about that upcoming vote there is a timing problem. They recommend phone calls.
Not necessarily. For example, I believe a big enough social media campaign could have convinced Dennis Kucinich to relocate to Washington State and become a rep there, instead of staying in Ohio and eventually losing his seat. (Of course, that never happened, and I never finished my version of what the OP made (which was basically Tweet-A-Senator, but you could only tweet Dennis Kucinich....)).
I think twitter can be used in innovative, non-traditional ways to influence politicians. #Benghazi, anyone?