If door seal is missing, can it trigger "CHECK VENT" light?
I have taken down all 4" duct work and cleaned it all. Have had dryer for over a year with no problems.
The door seal missing would not trigger a vent check light to come on. You will want to check the blower wheel to make sure that there is nothing caught in the blower wheel that would cause slow air flow. Also check the outlet thermistor to see if the issue would be caused by a faulty thermistor.
Does the adhesive pictured come with the seal?
Yes sir, the adhesive comes with the seal. If you need the adhesive but not the seal you can order it separately Y055980.