If I run a summer agenda making it summer and my opponent runs an agenda making it winter how does it resolve? If the opponent's card gives characters strength increases when it is winter and mine gives the same but in the summer...how does this work?

Can both run active at the same time? Can every player run their own choice of agendas even if some conflict? Does the agenda only give the benefit to the house it is attached to and not the opponent?

Thank you in advance!

Well, there are no Agendas that make it Summer or Winter, and the primary ways for making it Summer and Winter (the Black and White Ravens) discard each other as soon as they come into play. So this is a very unusual situation.

It can happen, though. If it does, then every effect that is "turned on" by Summer and every effect that is "turned on" by Winter are all active. At the same time. So, for example, the Kings of Summer Agenda would make you draw 1 more card (for Summer) and 1 less card (for Winter) at the same time - which means two effects of the agenda would cancel out if it happened to be both Summer and Winter at the same time.

Generally speaking, Agendas only affect the person whose House card they are attached to. Read the text of the specific agenda to be sure. Every player can run their own agenda. As for what to do when they conflict, I'd need a specific example because I can't think of any that do.