Does this stop something like Mirri Maz from killing the attached character? Does it also mean immune to Dany's debuff? Is this essentially a kind of Mini-Core-Catelyn?

Yes, if the attached character is participating in a challenge then it is protected from Mirri and also from Dany.

Note that Catelyn's Constant ability is slightly different - she prevents any card ability from triggering, so also from Attachments, Locations, Events, Plots or Agenda-rains - but only those which would need to be triggered -> bold precursor. And Dragonglass dagger makes a character immune to any character's abilities. So it doesn't protect vs Locations, Events etc. but it is preventing all abilities, not only those which would need to be triggered.

Thanks . Just to clarify - 'character ability' also covers keywords - like stealth? Keywords being abilities

It does cover keywords, but it won't protect you from stealth, because when stealth is declared, there are no defending characters yet, so the Dagger's ability is not active yet. Pretty much the only keyword it can protect from is intimidate, e.g. if you use Castle Black to stand your defending (i.e. participating) character.

mplain. Thanks. I overlooked the participating part.

So this protects against core Cait? If this is on a character that has a triggered ability, he can use it while Cait is in a challenge?

I don't think this protects from core Catelyn. Catelyn's ability prevents players from triggering card abilities. It doesn't actually target or directly affect cards.

While Catelyn is in a challenge she blocks opponent to trigger any ability. F.e. WotW Jon Snow. If he blocks Catelyn and wins he normally can't use his ability to stand all NW characters. But if Jon Snow has Dagger, he can use his ability normally.

Remember - while participating in a challenge, Dagger makes your own character immune to any opponent's character abilities, triggered or not. So also vs core Catelyn. Also bear in mind that Dagger protects only that particular character, it won't stop Catelyn blocking other triggered abilities you would like to trigger (like other characters, locations, events etc.).

I am not so sure about this. From stories I remember 1ed was full of such interactions. Catelyn does not target Jon, she targets you as a player. Jon is not impacted in any way, it is you who cannot trigger his ability. 

Catelyn Stark does not affect any characters with her ability. She affects players. So immunity to character abilities will not help the bearer of the Dagger against Cate.

If my opponent has Catelyn, Isn't her ability an "Opponent's character ability"? I'm confused. Please provide examples where Dagger works and where it won't. Maybe it will be more clear.

An example where it does work - if Old Forest Hunter (for example) has it they can count their strength when defending against Core Balon despite having lower strength. 

See the RRG entry for Immune:

Ok that is more clear. The Player can't trigger card abilities. How is it then with Forced reactions/interrupts? Are those considered to be triggered by player or by game? F.e. Hound if he successfully defends Catelyn, does he trigger his Forced reaction?

Also regarding Dagger - is there any other example where it won't work except Catelyn?

Forced Triggered abilities are not triggered by player (officialy), So Cat does not prevent them. 

So yes, Hound needs to have his reaction triggered.

What happens if attached character is the only choice after The Queen's Assassin is ambushed? Is he immune to that ability because it originates from an opponent's character, or does he die because the ability causes him to be targeted by his own controller rather than an opponent?

Being targeted by the opponent doesn't change anything - the character is still targeted by a character ability, and he is immune to it, sp the ability of The Queen's Assassin will do nothing.

What happens if attached character is the only choice after The Queen's Assassin is ambushed? Is he immune to that ability because it originates from an opponent's character, or does he die because the ability causes him to be targeted by his own controller rather than an opponent?

Unless the Ambushing is happening mid-challenge, the Dagger wielder is a legal target.