Shorten URLs for example like http://tinymdg.ccc/zxsddsd hard to find what is it relevant to even for real human eyes, so for search engines it's nothing no value for seo but for hiding the real URL or to make it short for special purpose.
I don't know if Twitter can help you with indexing, but I know for sure that Google does consider links coming from Twitter, Facebook, YouTube etc despite the fact they are nofollow. I can show you the screenshot of my Webmaster Tools account where it's clearly visible.
Why do you shorten your url's in twitter? In most cases 140 chars is enough to put whole url (at least for me) Anyway, using twitter to promote your url's is pretty sweet thing. You can see that seconds after you tweet your url bunch of bots visits your site. Probably indexing it too.
The people who say links on twitter don't count have no basis for their comments. I have hard evidence that suggests otherwise. (1) Google displays twitter links in google webmaster tools and properly parses / follows the shortened URL's, whether they're bit.ly or other URL shorteners (I'm not sure if there are any obscure ones it does not follow/understand), which proves that google has the capacity to understand and follow the links. (2) I've seen evidence from my own sites that new pages that I tweet and that are retweeted by a large number of users tend to be indexed more quickly than ones that I tweet and are not retweeted, and these pages tend to be indexed more quickly than ones that I don't tweet. I don't know if it's a direct effect -- it could be an indirect effect of people adding links to these pages on other sites when they find them on twitter -- but I suspect that google takes twitter and other social networking shares into account in indexing and possibly even search ranking. They'd be silly not to: these things are a good indicator of link popularity. Don't just pull reasoning out of your rear end.
The thing you have to remember is there are many sites that will scrape your "tweets" and post them on their site. Like friendfeed for example.. So to answer your question, yet posting your links on twitter will help your site get indexed, but this shouldn't be the only thing you do since those updates will quickly be pushed to the 2nd, and 3rd page. Keep posting quality content, and your site will get indexed.
Yes quality is essential. I personally experienced the power of twitter and some other social network sites like digg and reddit for fast indexing of any webpage.