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Yes and you are better off to do it immediately--waiting a number of years makes no sense. They obviously liked you if you made it that far into the hiring process, and should remember you still, which could streamline your application.
Personally I would send it in through whatever channel they are offering, but if you met with a recruiter in HR or a hiring manager, drop them an email expressing your interest in the position and attach your resume and a fresh cover letter.
Yes, absolutely. If you got that far the other time around, they probably really like you. You may have been rejected only because they had to chose between a few excellent candidates. I have been in the position before of having a number of candidates that I would love to have hired but only one position. If another opened up, I would have been happy to hire one of the people I was forced to reject for the first job. Don't let your previous rejection leave a bitter taste in your mouth. If you want the job, go for it!