Since J. P. Johnston has covered just about everthing else I'll confine my response to your questions regarding unit designation and transportation.
Quite simply, armored infantry receives specialized training in tactics involving the support and protection of tanks and also in penetrating or circumventing an enemy tank's defenses. This may sound odd, but tanks, although designed to be mutually supporting, have "blind spots" that can be exploited by individuals daring and determined enough to do so.
Armored infantrymen were transported by vehicle - trucks, half-tracks and even tanks - whenever possible. Usually on foot, in tactical formation, only while in the attack.