Well, all of your questions are really answered by the rules:
Q1. Does a character need to have an intrigue icon in order to defend against an intrigue challenge, or can the defend with anyone?
A1: Yes, the character must have the intrigue icon. Any character that a player wants to declare as a defender in a challenge must have the challenge icon for that challenge. (Rule Book, p. 14: "The opponent you are challenging now has the option to kneel any number of his or her characters that have the corresponding challenge icon (or are enabled to participate by some card effect) to defend against your challenge.")
Q2: Does a player have to have characters with the icon for the challenge type in order for me to challenge them in the first place?
A2: No. You can attack anyone, whether they have characters that can be declared as defenders or not. Nowhere in the rules will you find something that limits a player's ability to attack based on another player's ability to defend.
Q3: Can I defend with a character that is already kneeling?
A3: No. Characters must be standing to be declared as a defender. Characters must be both standing and have the appropriate challenge icon to be eligible as defenders in a challenge. (Rule Book, p. 14: "Already kneeling characters may not be declared as defenders.")
NOTE: There is a sentence on p. 14 of the rule book that confuses a lot of people. It reads, "Your opponent must declare at least 1 defending character in order to be considered defending against a challenge." This statement is there to remind players that there is a consequence to not declaring defenders (the bonus power to the attacker for winning a challenge "unopposed"). But it does not contradict the fact that the defender is given the option to declare any number (including 0) of their eligible characters as defenders. Further, this statement does not limit the attacker to picking defenders that have the option to defend; if it did, it would also give the defender the option to completely negate the challenge by not declaring any of their eligible defenders, and in Westeros, the advantage tends to be with the aggressor.
Also note that the above answers are the basic, general answers. Different card effects can always exist that say Player A cannot attack Player B, or that a character without the icon or that is kneeling can be declared as a defender. Just know that a card must specifically say that something other than the above answers is true before you do something else.
So to summarize the issue:
1. Attacking players don't need to choose opponents who have the ability to declare defenders; they can go after anyone.
2. Defending players can only declare characters who are both standing and have the appropriate icon as defenders.
3. Even if the defending player has eligible defending characters (i.e., standing and with the icon), they can choose to declare no defenders at all. They'll probably lose the challenge and the attacker will get the "unopposed" bonus, but they can choose not to defend.
I'm not sure that these rules will be made any clearer by naming cards with and without the intrigue icon?