Submission procedure for letters of recommendation
Consult the application page of the school you are applying to. Most institutions make it pretty clear how letters of recommendation are to be submitted. From an ethical standpoint, the letters should never pass through your hands, just as your transcript should be mailed directly from your undergraduate institution without going through you.

If the application process is completed online, as most are these days, you will have to provide email contacts for your references. They will receive emails from the application system on how to proceed. Their access to the system should be generated automatically. Neither you nor they should have to create accounts for them.

If the institution is insistent on using snail mail as the means of submitting information, then provide each of your letter writers with an envelope that is already addresses and has proper postage. That way your letter writers can seal the envelope themselves and drop the letter in the mail. 

It is possible to submit the same letter to another institution. I have read some letters that were clearly done that way (e.g., addressed to a different institution, describing the candidate's aptitude for an unrelated position, photocopied, or dated five years ago). I would not do this. Submitting identical letters to multiple institutions (and it is usually obvious, because a good letter is tailored to that institution), implies three bad things about you:


You are too lazy to ask for new letters of recommendation.
Your letter writes did not know about the second application, so their comments are not relevant.
You didn't trust your letter writers to write you a good letter for that position.


Just ask your letter writers to write new letters to each institution. This second letter will not be written from scratch, so it won't be as much work for them. Likely, they will edit a letter they wrote about you previously to create a new letter specific to the new institution or position.