Any Phase: Kneel The Scourge to choose 1 character. That character loses a Military, an Intrigue, and a Power icon until the end of the phase.
Response: After a character loses an icon from The Scourge, choose and discard 1 card from your hand to have that character gain that icon until the end of the phase. Any player may trigger this effect.

The first question is for Joust games.

I use The Scourge on an opponent character. How many times can the Response be activated? I assume only once. It means only one icon can be given back.

For Melee, it would be the same then. Only one player can activate this - the first who wants to activate this repsonse. And then only one icon will be back.

Because if this Response can be activated multiple times, then it is possible to give multiple Military icons by one player?

Or one player can give back the same type of icon only once, but in Melee every player can give back each icon type once? So in a 4 player Melee - assuming that I do not want to give back icon to tha character - that character can have -1+3 Military -1+3 Intrigue -1+3 Power icon?

The response can be triggered as many times as icons lost from the location, meaning a max of three (1 of each type). For each icon lost, you can discard a card to gain the same icon back. However, you cannot give the character icons that it did not originally have. For example, if The Scourge was used on a character with only a military icon, you cannot trigger the response and give it and intrigue icon, since The Scourge never removed that icon from the character. You can't give multiples of the same icon either, for the same reason.

Once per icon lost (so, as Crevic says, a maximum of 3 times). The "one Response per trigger" rule applies, but there are 3 triggers created - one for each icon.

The number of players in the game has nothing to do with it. The "one Response per trigger" rule specifically talks about one Response per trigger per card, not one Response per trigger per player. The first player to trigger the Response for each icon covers the limit.

When you are triggering the response to gain back lost icons, can it be done later in the round or do the cards have to be discarded at the time of removal?

thanks in advance.

At the time. Its a response, like any other, and it has a window in the player action for the opponent to choose to discard or not, if they pass the window closes and the option dosent come up again (unless it is triggered again, but that would likely be targeting another character so you could respond to the new target having icons removed).

Hope that helps