QI have an Orchis, Vengeful Puppet and Puppet token on my field. If I attack an enemy Rookie Succubus with my Puppet token, causing both to be destroyed, can I select the enemy Forest Bat token summoned by Rookie Succubus's Last Words ability and deal it 2 damage?

ANo. When both players have pending abilities, the active player must resolve theirs first. In this case, you are the active player, so you must resolve your Orchis, Vengeful Puppet's automatic ability before the opponent can resolve their Rookie Succubus's Last Words ability, and you would need to select a target to deal 2 damage to before the enemy Forest Bat token is summoned.

QI have an Orchis, Vengeful Puppet and Puppet token on my field. If I play Nature's Guidance and select my Puppet token as its target, do I deal 2 damage to an enemy leader or follower?

AYes, you do.

QI have an Orchis, Vengeful Puppet and 4 Puppet tokens on my field. If an opponent plays Themis's Decree and destroys all of my Puppet tokens, do I deal 2 damage to an enemy leader or enemy follower on the field 4 times?

AYes, you do.

QI have an Orchis, Resolute Puppet and 2 Puppet tokens on my field. If an opponent plays the Quick spell Angelic Barrage during my end phase and destroys both of my Puppet tokens, do I put 2 Lloyd tokens into my EX area?

ANo, you would only put 1 Lloyd token into your EX area.

QI have an Orchis, Resolute Puppet and Puppet token on my field. If I play Nature's Guidance and select my Puppet token as its target, do I put a Lloyd token into my EX area?

AYes, as long it's the first time a Puppet token has left your field that turn.