Making an appointment regarding letters of recommendation
This may vary between departments, countries (I'm in the U.S.), etc., but I'd say:


When making an appointment, it's almost always a good idea to specify what it will be about.  When someone doesn't have a lot of time, this helps them figure out how soon the meeting needs to be scheduled, and it gives them a chance to prepare for it.  The one exception I can think of is very delicate topics.  For example, if you are meeting with someone to discuss an ethical matter, there might not be a short description you can give that wouldn't risk be misleading or omitting important context.
I'm not aware of a belief that asking for a letter of recommendation via e-mail is rude or problematic (although of course this may depend on the local culture).  In fact, I would prefer to be asked by e-mail.  Partly it's just because I generally prefer e-mail for things that can be handled that way, since it's less disruptive.  Partly it's because awkward situations are a little easier for me to handle by e-mail.  For example, if a student didn't do well in my course, I may point this out and ask whether there's someone else who could write a more compelling letter.  It's easier for me to find the right words in writing, rather than on the fly in a conversation.