… Unless the answer is no, in which case I'm glad I asked it. If you have a character agenda (Griff, Quentyn, etc.) attached to your house card, are you "running an agenda" for purposes of cards like Heavy Taxes or Seal of the Crown?

character agendas are still agendas, the attachments will stop working while the character is attached to your house

Yup, they count as agendas, although they are a specific subset of agendas known as "Character Agendas". The only difference is that there are a few cards that specifically refer to character agendas (eg http://www.agotcards.org/card/v/4505 ), which in that respect treat them differently from other agendas.

In every other sense they are agendas - they are out of play (so you can play another copy of even a unique character), they cannot be blanked, they affect Northern Cavalry Flank & Heavy taxes.

Thats what I thought - thanks!