If the targeted Forest entered this turn, it will be affected by "summoning sickness" once it becomes a Treefolk. You won't be able to attack with it or use its activated abilities that have {T} in the cost (including its mana ability).

Awakener Druid's ability is mandatory. When it enters, you must target a Forest (if there is one), even if you don't control that Forest.

Awakener Druid's ability affects a land with the land type Forest, not necessarily a land with the name Forest.

If Awakener Druid and the Treefolk are dealt lethal damage at the same time, both will be destroyed. However, if Awakener Druid leaves the battlefield before the Treefolk is dealt damage, it will immediately revert to being just a land and thus can't be dealt damage.

If Awakener Druid leaves the battlefield before its "enters" ability resolves, nothing happens to the targeted Forest when that ability resolves. It won't become a creature.