Is it possible for your air conditioner and heater to blow at the same time?

Probably not but it can switch from heat to ac if this answers your question

First responder is correct, if you have a system know as a heat pump. If you have separate heat and A/C , having you A/C turned to extreme cool may cause your heat to come on. A window unit blowing hot and cold simultaneously is indicative of the A/C motor overheating, and could be dangerous.