If Jon Snow is given stealth, does his ability actually trigger the Stealth? I'm not convinced because of the distinction between declaring and participating. Here are the necessary rules:

Jon's ability says he is "...considered to be participating ..."

Stealth says "...for each character [the player] declares as an attacker..."

The rules for participating make it clear that if you declare, you are participating. But I'm not sure it goes both ways. What do you all think?

You are correct. All characters that are "declared" in a challenge are considered "participating," but not all "participating" characters are declared.

Characters that jump into challenges using card abilities like Jon's are not considered "declared," and so would not activate anything that required them to be declared (like Stealth).

However, there is another side to this. Anything that limits the number of characters that can be "declared" (like the plot Jousting Contest) will not stop a character like Jon from participating in a challenge through a card ability without being declared.