Robert Baratheon enters a military challenge with Starks, because of the number of knelt characters involved Robert has more strength, Starks are able to sacrifice a character to stand everyone knelt taking the strength from Robert and winning the challenge. Is this correct.

Stark has a plot card that allows a stark to participate in a challenge, can this be a a character with out the icon for that challenge type.

1) Yes, as long as the sacrificed character wasn't contributing his Strength to the challenge.

2) Yes; challenge icons are only necessary for being declared as an attacker or defender. Effects that say to add someone to a challenge ignore icons (Ours Is The Fury, Winterfell Kennel Master).

For 1): You'll have to work out the math in all cases, but it's possible whether or not the sacrificed character was contributing his STR.

1. Point is, Robert's ability is continuous, so his STR constantly changes - in both directions - as characters kneel and stand. If characters kneel or stand before challenge resolution, it will affect the STR he contributes when determining who wins and loses the challenge.

Thanks

When Robert Baratheon attacks, is his strength is set at the time he is declared an attacker (i.e. five plus one for any other characters knelt at that time) or does he gain additional strength when the challenge is in progress due to the characters the opponent kneels to defend?

Robert's strength is continuously updated. As the Rules Reference says:

Modifiers
Some abilities may ask players to modify values. The game state constantly checks and (if necessary) updates the count of any variable quantity that is being modified.

As stated above, Robert's STR is constantly changing based on the numbers of standing and kneeling characters. The STR he contributes to the challenge will depend on the number of kneeling characters when the challenge is resolved, not on the number of characters kneeling when he is declared as an attacker or defender.