How would first strike and deathtouch interact with multiple blockers in this scenario?
You would only be able to destroy two of the blocking creatures.  For deathtouch to have an effect, you must do damage to a creature. On the plus side, an attacker with deathtouch only needs to assign one damage to each blocker.  In your scenario, your creature is only a 2/2, and so even with a team of blockers, only two damage can be shared among them.
Based on your scenario's here is the change needed to leave your creature alive yet destroy all blocking or attacking creatures: your Entangler must be at least a 5/1.  I say a 5/1 because toughness can't be 0 and you must be able to deal at least 1 damage to all blocked or blocking creatures. Using a card such as Giant Growth would allow your creature to live assuming it has both first strike and death touch.  Without the first strike as long as you have 1 power for all attacking or blocking creatures you would be able to take them out as well.


702.2b Any nonzero amount of combat damage assigned to a creature by a source with deathtouch is considered to be lethal damage, regardless of that creature’s toughness. See rules 510.1c–d.
704.5h If a creature has toughness greater than 0, and it’s been dealt damage by a source with deathtouch since the last time state-based actions were checked, that creature is destroyed. Regeneration can replace this event.