When I try to preview certain sounds or play them in-game... they won't play. I can't hear them. And even if I restart Hammer or refresh sounds, etc. I can never hear them. Specifically sounds where people talk or sounds like this: announcer.OpeningExercise01 or 02, 03, so on and so forth. Does anyone know how to fix this?

You are using soundscripts for an actor which cannot be heard inside Hammer but should be able to be heard within the game.
You can play them through ambient_generic and should have the Start Silent and Not Looping flag ticked.

Looking at the soundscript data: it doesn't have any audio emitter points so you can play it straight from the origin of an ambient_generic. Alternatively if you still cannot get it to work: you can play it from the raw file which is vo/announcer/OpeningExercise01.wav.

"announcer.OpeningExercise01"
{
   "channel"   "CHAN_VOICE"
   "volume"   "VOL_NORM"
   "pitch"   "100"
   "soundlevel"   "SNDLVL_NONE"
   "wave"   "vo/announcer/OpeningExercise01.wav"
   "soundentry_version" "2"
   "operator_stacks"
   {
      "update_stack"
      {
         "import_stack" "update_dialog"
         "mixer"
         {
            "mixgroup" "announcerVO"
         }
      }
   }
}

It worked, thank-you.