This thing hasn't got so much to do with warmup and cold weather per se; So yes, you need this tube all the time if you want max mpg.
1) The door thing on the air cleaner generally works all the time---not just during warmup phase.
a) You could be driving in the middle of summer in arizona, and the door will still be partially open, unless the outside temp gets higher than 117 degrees which is unlikely.
2) Warmer-----LEANER air = less fuel needed to provide combustion= better mpg.
a) The door thing is designed to provide intake air of 84ish to 117ish degrees (will double check when link provided) ALL THE TIME.
3) Yes colder air----denser air is better for performance. More fuel burned with the air=more power.
Sorry guys, but the GM engineers have written books about this; i will provide the links when i get more time. i am not an engine person, but this is a mis-conception and we should clear it up.
The only reason why i know about this is because weird stuff like this fascinates me; Next step----vacuum hoses!