This happened to me and enough people disagreed with me I let it happen, since I didn't understand the ability well enough to know, myself. If a opponent casts Act of Treason on my Kotose, the Silent Spider, can they then cast the card I previously exiled with her triggered ability?
Good news: Your opponent cannot cast an exiled card. The entire effect is part of her ETB triggered ability, so changing control of Kotose doesn't suddenly allow someone else the ability to cast it. (It would need to be a separate ability on Kotose for that.)
Bad news: You can't cast the card either anymore, since you've now lost control of Kotose and even when you get her back in the cleanup step, the duration for her last clause has lapsed and can't be restored for any card she has already exiled.
