You can use absolutely everyone. You bring a 20-person squad with you into each mission as your selection of deployables (Although... five? Of those are fixed), with you rarely ever using every single soldier in the squad on a given stage. It's just the list you have immediately available. However, you're able to freely change your squad's composition between stages. In addition, you level up the classes, not the characters - so if you have level 10 snipers, all five possible snipers function at level 10.
(Of course, two characters don't necessarily function the same. The individual potentials and personality traits can make two characters in the same class perform very differently in the same circumstance. Some characters are ultimately better than others, although what level they're at, and other factors (such as if you've unlocked/upgraded their final personality trait through repeated use) also come into play)
Past the main characters who join at fixed points, you'll get a random selection of characters to choose from at the start, and the over the course of the game the remainder of the random pool will join. There's also one character for each of the five classes that is conditional to appear and join. Yes, a couple of these require specific characters to be wounded in battle. Not killed - just wounded (Reduced to 0 HP, but then saved either by having another soldier reach them within 3 turns or finishing the battle in that time)
