Text on Robert is:

"While Robert Baratheon is attacking alone in a POW challenge, each opponent you are not attaching may declare 1 eligible character as a defender."

The card does not specify anything about when or how those defenders are declared, so they would be declared as part of the standard "declare defenders" framework event. That event initiates with the defending player declaring (non-Stealthed) characters as defenders, if they so choose. Since the addition of extra players getting to declare defenders as specified by Robert's text becomes part of this framework event, which starts with the defender, the "extra defenders" will be declared clockwise from the defender (who starts that framework event).

So if Robert is given Stealth, can he use it on a character controlled by a player other than the one he has just declared his challenge against? The Core Set rulebook defines eligible Stealth targets as "a character without Stealth on the defending side." With Robert, what does "defending side" mean? Is it every opponent's characters?

I don't think so, because I don't think they are considered a defending player. If they were innate defending characters the second part of Robert's ability (which says take power from each opponent) would be assumed, and not on the card as a replacement effect for normal claim. Since there is a replacement effect, it isn't normal claim, and they are not defending.

That's my reasoning, feel free to use it as a chew toy.

Correct. The "defending side" is still the player the challenge was declared against - not every other player. Robert's effect allows the "uninvolved" players declare a defender and requires them to fulfill claim when the attacker wins, but it does not make them "defending players," nor does it include them on the "defending side."