I know that you're supposed to sing from your diaphragm, but are we normally supposed to speak from it too?

Singing and talking require different amounts of air. Singing one word can take more breath than speaking an entire long sentence. Actors, public speakers, do use their diaphragms so they can project their voices or make them sound more resonant--but I'm a singer and I know I don't speak as I should because my voice is tired out after a day's work where I might have had to talk nonstop. Of course, I've never had to sing for 8 or 10 hours straight with one half hour lunch break. I've had all day rehearsals for things, but there was a significant part of downtime. There are speech and voice therapists that occasionally answer singing-related questions and maybe one of them will also answer your question.

In fact, yes, especially if when working as a teacher/telemarketer/actor or in any occupation where one has to talk aloud a lot. This is called "vocal support" and it takes away some of the load of the vocal chords. By using the diaphragm properly one does not need to strain the vocal chords too much, and too much straininess is harmful for your throat (think of vocal nodules, hoarseness and other nasties).

That is called projection and everybody who speaks in public has to learn it. You do it any time you need to. If you don't, people just won't hear you.

In fact, yes, especially if when working as a teacher/telemarketer/actor or in any occupation where one has to talk aloud a lot. This is called "vocal support" and it takes away some of the load of the vocal chords. By using the diaphragm properly one does not need to strain the vocal chords too much, and too much straininess is harmful for your throat (think of vocal nodules, hoarseness and other nasties).

That is called projection and everybody who speaks in public has to learn it. You do it any time you need to. If you don't, people just won't hear you.