recommendation letters at different stages of interviews
What to do to avoid requesting letters too often, annoying letter writers?


You don't request letters “too often”.  You request letters once, at the beginning of your job hunt: “Would you be willing to write a strong recommendation letter to support my applications for [kind of job]?  I'm planning to send out about [number] applications.”  If they say yes, take them at their word.

At least in North America, academic/research employers request letters directly from your references, not from applicants.  Applicants only provides names and contact information for their references; they never see the actual letters.  Most universities send letter requests by email; a few ask references to submit their letters when the applicant submits their other materials, without waiting to be being asked directly.

Keep your references up to date on which [kind of job]s you've actually applied for, and how they should expect to receive the letter request.  I strongly recommend maintaining a single Google spreadsheet (or your favorite equivalent) with a row for each target institution, and columns for each of your references to indicate who has asked them for letters, and which letters they've actually sent.  (Then your references can see which of your other references have already submitted letters, making it less likely that they'll fall behind.  Peer pressure works.)