I doubt many of these terms have very precise definitions, and I am probably not the best authority on their meaning, but here is what I would take these terms to mean:
Serious fiction: Not genre (bearing in mind that genre is defined no only by subject matter but by treatment and intent -- Cormac McCarthy is not genre western for instance).
High quality fiction or quality fiction: Not genre.
Handles all subrights of book (what does that all include typically?) Movie, ebook, translation, foreign, adaptations. Basically all the various places and form your copyright can be licensed to.
Full service literary agency (what is full service?) Handles all of the author's literary affairs.
Handles all subsidiary rights (what does that all include typically?) subsidiary rights = subrights
Copying/photocopying fees (for what? and how much typically?) You pay when they make copies to send out. Probably at cost, but your should ask.
Only accept work from Professional writers (who are professional writers?) May mean writers with existing paid credits. May mean writers who are intending to pursue a full time literary career. May mean writers who are already making a living writing. There is not much money in one-off books.
Accept by referral only (what kind of referrals?): Someone they know and respect recommends you to them. Might be an editor, one of their current clients, a writing teacher, another agent who things they might be a better fit for your work. It may also means people they have met at pitch sessions at writing conferences. Basically means they are not reading the slush pile.