A: Yes, you'll still need a resume.

As you've evidenced through your friends, social media--including Twitter--have changed the way job seekers seek and recruiters recruit. More and more HR departments are using Twitter to reach out to prospective candidates, either in lieu of or in addition to more traditional methods of recruiting. And job seekers are growing more savvy, too.

Keep in mind, though, that Twitter is being used a tool of introduction. At best, some skillful, insightful use of one's Twitter account might lead to a conversation and an interview. It isn't a job application form, and it isn't taking the place of a resume. At some point, each of the people you know who've gotten jobs through the use of Twitter were likely required to submit a resume. It may have come sooner (after an initial exchange of tweets) or later (at or subsequent to a face-to-face meeting). Regardless, the hiring official or hiring committee is going to want to see your qualifications on paper before extending a job offer. And the more impressive the marketing of your brand (on that paper), the greater the likelihood of that offer.  

By all means, explore the use of Twitter as a potential tool in your job search. But keep a well-crafted resume in your proverbial hip pocket, ready to employ when called for.

BTW, don't approach a unacquainted hiring manager, ask for a job in a tweet, and expect to get any kind of positive response. A recent article from Kiplinger's Personal Finance suggests that you "Aim for a genuine relationship before respectfully soliciting advice or requesting a face-to-face meeting." -- How To Use Twitter To Help Land A Job, July 2013.

Good advice.