Best extraction fan with compatible CFM for grow tent 4ft x 4ft x 6f .t

AC Infinity Cloudline all the way, probably the 6". Spend the money to get the model with the programmable controller so the fan will turn on/off as needed to maintain the temperature and humidity. I have a 3x3 tent and use a 4" Cloudline and the highest I have ever set it is level 5 (half power) but most of the time I use level 3. In a 4x4 tent you could still use the 4" fan but would need to run at a higher setting which increases the noise. At level 5 or lower I can't hear my fans running unless I get close to the blower. A 6" fan can be run at a lower setting than the 4" so it is super quiet and would have extra power in reserve in case you need it.

Yes, another AC Infinity T6 person here. Works really well in a passive setup, but I also have a S6 now as an intake fan, set way lower speed than the T6 to keep negative tent pressure. Honestly has more firepower and CFM at top speed than you'll need in a 4x4 but running it down in the 5-7 range is efficient and quiet, even with a filter on the front end.

I would go with an AC infinity T4 or T6. Having the T means it will come with a humidity/temp sensor that you can configure to turn on based on desired ranges. It works great for climate control. The T4 is a 4" and should be good enough, But I would recommend a T6 as it will require less effort and your carbon filter will work better.

I use a 6” AC infinity. Though a 5” fan is more than enough to get air moving. I’m not really sure what your question is. I watched the video and your set up seems fine. That oscillating fan I would hang that upside down but other than that looks fine to me.