QIf my leader has 1 defense, and I give my leader -1 defense to play this follower's activated ability, but I select a Demon follower that originally costs 2 or less play points in my cemetery and put it onto my field, do I lose the game?

AYes. After you finish resolving this follower's activated ability, Confirmation Timing occurs, during which loss conditions are checked. Since your leader's defense is 0 at this point, you lose the game. In other words, you check whether or not you've lost the game before you give your leader +1 defense with this follower's automatic ability.

QIf I put 2 Demon followers onto my field at the same time—with Demonlord Eachtar's Fanfare ability, for example—does this follower's ability trigger 2 times?

AYes. It triggers twice, so you give your leader +2 defense.