What happens with Call to Battle and Armored Tusker to a battlefield?

Yes — moving Armored Tusker with Call to Battle is a genuine Move, so its "When I move" ability triggers and deals 3 damage to a unit at a battlefield (that's its Shield value of 3, not its Might, and not the +3 it gets while defending).

Sequencing is the interesting part: Call to Battle resolves completely first — the Tusker moves, then your opponent is forced to move a unit to that same battlefield — and only after that does the pending move-trigger go on the Chain (CR 455). That means you can point the 3 damage at the unit your opponent just moved in, and anything at 3 Might or less dies before Combat opens.

Two conditions to keep straight: the destination must be a battlefield you control (FAQ #11706), and the Tusker's own move doesn't apply Contested since you already control it (CR 452). Your opponent's move does apply Contested, so a Combat is staged where they're the Attacker (CR 466.2.c.1) and Tusker defends with Shield 3 — 7 Might, and Tank means it takes lethal damage first if it survives the trigger.