We're new to the game, and haven't really delved into the other forum but had a few questions about renown.

The Power earned from renown is kept on the card?

You can earn the Power from renown even on defense?

What happens to the Power on the Character if it is removed from play or killed?

The Power earned from renown is kept on the card?

You can earn the Power from renown even on defense?

What happens to the Power on the Character if it is removed from play or killed?

Yes the power is kept on the Card.

You can earn power even on the defense, the character has to be participating in a challenge and win doesn't matter if its on the attack.

if the character is removed from play the power is lost. this can be bounced to hand or discarded or killed, anything that removes it from play results in the renown being lost. Don't forget that effects that "Save" a character such as duplicates will stop a character from being removed from play and thus it will keep its power.

Also keep in mind that if you have no power on your faction card, your opponent cannot take power from your character cards if they beat you in a POW challenge. So there are risks and benefits.

Sort of related question: when a card (like Lightbringer) says, "After attached character gains 1 or more Power...," that's referring solely to something like Renown or Consolidation of Power, right? Not from earning you Power via a challenge?

You are 100% correct that it's when an effect has you take a power token from the power token pool and has you place it on a character. In this case, it needs to be the attached character that gains power in that way.

When you place power on your faction card, it's not a character gaining the power.