With "The Siege of Winterfell" Agenda, does this mean that while it is running that player cannot claim power for Dominance or through Event cards?

That is correct. You could claim power through event cards during a military though. Like make an example could be used on a military challenge, but you could never use Superior Claim because it only works on a power challenge.

Makes sense, thanks a lot.

On a power challenge, you can still "move" power for claim since "moving" the power is not "claiming" power for your house. You still cannot claim an unopposed power for power challenges. You didn't directly ask this, but it is a common early mistake with people new to this agenda.