When I triggered an effect for example using the Iron throne on a character. If the opponent then gives it immunity through mereen but after I triggered my effect on it. Does this immunity cancels my triggered effect?

In this case, definitely not. Iron Throne's character ability cancel happens during the save/cancel step after a character triggers their ability. Meereen is a player action played during challenges in its own Player Action Window. By the time they could kneel Meereen to make the character immune to triggered effects, the cancel would have already occurred.

When I triggered an effect for example using the Iron throne on a character. If the opponent then gives it immunity through mereen but after I triggered my effect on it. Does this immunity cancels my triggered effect?

No.

The Iron Throne completely resolves before the immunity is given to the Dragon character.

In addition to that, "immune to triggered effects" will not protect a character's triggered effect from being canceled by another triggered effect because the scope of immunity only protects the character itself. What it WOULD protect the Dragon character from is the "Then, kneel that character, if able." part of The Iron Throne because that specifically affects the character and not the the character ability.

Generally, you cannot be "retroactively" immune to something. Meereen will not "erase" triggered effects that have already resolved - even if they have lasting effects, such as "loses an icon until the end of the round" - when the immunity is gained.

Think of it like this, if you have chicken pox, then get injected with the vaccine, will your chicken pox just disappear?

When I triggered an effect for example using the Iron throne on a character. If the opponent then gives it immunity through mereen but after I triggered my effect on it. Does this immunity cancels my triggered effect?

Just some clarification on The Iron Throne and Meereen.

Meereen - " Challenges: Kneel Mereen to choose 1 participating Dragon character you control. Until the end of the phase, that character gains 'Immune to opponents' trigggered effects.' If you win the challenge draw 1 card for each participating Dragon character you control. "

The Iron Throne (Lannister) - " If you are the only player to control The Iron Throne it gains: 'Response: Kneel The Iron Throne to cancel a character ability just triggered. Then kneel that character, if able. "

The Iron Throne is a response (and more specifically a cancel-type response) and Meereen is a Player Action. Only cancel-type responses can interrupt the resolution of another action. As Meereen is not a cancel-type response (nor is it a response at all) it can not interrupt and cancel The Iron Throne in the way the game defines cancel.

Re: Meereen, I was playing against a Targ dragon deck two days ago, and the Targ player used Meereen during a dragon attack, then drew a card. So far so good.

Then he found a way to stand the dragon, and won another challenge. He went to draw another card from Meereen, but I said he couldn't.

His position was that it says "Until end of phase", while mine was that the card drawing was part of a different sentence to that phrase, so was a one off effect that only counted during the challenge that Meereen was activated.

He deferred to me on this, but I wanted to check if that was correct?

You were correct; the "If you win the challenge" refers only to the challenge in which the dragon was participating at the time you used Mereen. The immunity, however, is until end of phase.