QIf this follower is on my field, and I attack an enemy Lævateinn Dragon, Defense Form with a 1-attack Onion Patch, how much damage does Lævateinn Dragon, Defense Form take?

AIt takes either 0 damage or 1 damage. When there are multiple abilities modifying damage dealt, the player receiving the damage decides the order the abilities are applied. In this case, depending on the order, the final damage taken can be calculated as follows:

Lævateinn Dragon, Defense Form takes 1 damage because its controller decides to apply this follower's ability to increase the damage dealt by 1 first, so the Onion Patch will deal 2 damage before Lævateinn Dragon, Defense Form's ability reduces the damage dealt by 1.

Lævateinn Dragon, Defense Form takes 0 damage because its controller decides to apply Lævateinn Dragon, Defense Form's ability to reduce damage dealt by 1 first, so the Onion Patch will deal 0 damage before this follower's ability increases the damage dealt by 1. However, this follower's ability only takes effect when damage is dealt, and 0 damage doesn't count as dealing damage, so the ability doesn't take effect and the damage taken by Lævateinn Dragon, Defense Form remains 0.

QIf this follower is on my field, and I attack an enemy leader with a 0-attack Onion Patch, does it deal 1 damage?

ANo. When a 0-attack follower attacks, it doesn't count as dealing damage, and this follower's "if an Onion Patch on your field would deal damage, it deals that much plus 1 instead" only takes effect when damage is dealt.

QWhen does this follower's "if an Onion Patch on your field would deal damage, it deals that much plus 1 instead" ability take effect?

AIt takes effect when an Onion Patch on your field deals combat or ability damage. Note that when a follower with 0 or less attack deals attack or combat damage, no damage is dealt at all (it doesn't count as dealing damage).

QDoes Bergent, Onion Patchmaster's "if an Onion Patch on your field would deal damage, it deals that much plus 1 instead" ability increase damage dealt by Onion Patch's "Strike - Select an enemy follower on the field and deal it 1 damage" ability as well?

AYes, it does.