QIf I play this follower's On Evolve ability when my opponent has a Relic Goddess on their field, do they summon a Lococo’s Teddy Bear, even though Relic Goddess isn't destroyed?

ANo, they don't.

QIf my opponent plays this follower's On Evolve ability when I have 5 followers on my field, including a Mordecai the Duelist, can I play Mordecai's Last Words ability to summon it again?

ANo. This follower's On Evolve ability would cause you to summon 5 copies of Lococo’s Teddy Bear first, leaving no room for a Mordecai the Duelist, even if you play its Last Words ability.

QIf my opponent plays this follower's On Evolve ability when my field consists of a Greatpick Corpse and 4 copies of Forest Bat, can I summon a Forest Bat token with Greatpick Corpse's "when a 1-cost token card you control leaves the field" ability?

ANo. This follower's On Evolve ability would cause you to summon 5 copies of Lococo's Teddy Bear first, leaving no room for a Forest Bat token.