I ran across a couple of quests (Ruins of Belegost and Ring Goes South) where an enemy pops up and is "considered to be engaged" with both players, or one player, or whatever. I'm not quite sure what "considered to be engaged" means precisely. Is it engaged? What I am gathering is that it stays in the staging area, but it attacks players that it is "considered to be engaged" with and those players can attack back without having special staging-area attack cards, mostly.

But there are a few effects that I just don't know if they trigger or not. Haldir is one, if an enemy shows up that is "considered to be engaged" with Haldir's player, can Haldir use his combat action on that turn? Or not?

Same question really with a lot of the rangers, Mablung, Leader Faramir etc, do they trigger on something that is "considered to be engaged with" the player?

Pretty positive he can use his ability (Haldir) as you did not "engage" an enemy there is simply an enemy in play that is "considered" to be engaged with each player (more so for attack purposes and so it can stay in staging and contribute its threat than engagement purposes). Not 100% but pretty sure this is the case.

As for the rangers I am assuming no because you did not truly engage an enemy you are simply considered to be engaged for other purposes (again so it can both attack you and stay in staging).

I think with Haldir you can use his ability (because you have not actually engaged an enemy that round) but you can't use it against the enemy you are "considered to be engaged with".

Support: the FAQ

(1.50) “Considered to be engaged” vs actual engagement

An enemy that does not leave the staging area but is considered to be engaged with a player does not actually engage that player, nor does that player engage it. In order for a player to engage an enemy, the enemy card must physically enter his play area.

Mablung definitely doesn't work - this is also covered in the FAQ:

For example: Durin’s Bane (D 150) cannot leave the staging area and is considered to be engaged with two players. Player 1 has Mablung (RM 84) and wishes to trigger his Response effect, but he cannot because he has not actually engaged Durin’s Bane. Player 2 wishes to play Feint (CORE 34) to prevent Durin’s Bane from attacking him. He can, because Durin’s Bane is considered to be engaged with him.

Yeah, since it has been confirmed that Mablung and similar effects don't work for enemies considered to be engaged, logically it follows that Haldir does still work in such circumstances.

And how about when it is 'put into play this enemy engaged with you'? Haldir, Mablung, Sam, etc triggers or not?

^ that was my next question

Yes. That one's even specifically in in the FAQ I think. An enemy which is put into play engaged with a player has engaged that player.