Can someone help me analyze this article?
I will answer the questions one by one-


Why does this miniaturization necessitates insulation ?
An organism's volume determines the total amount of heat that can be stored. The loss (exchange) of heat between the body and external environment mainly occurs on the skin's surface. Hence, body volume determines how much heat is stored, while body surface determines how fast that heat is dissipated to the environment. Volume increases with a power of three with radius, while surface increases with a power of two. Hence, smaller animals have large surface-to-volume ratios, which decreases rapidly with body size. Hence, small animals will dissipate relatively more heat per unit of time.
Dinosaurs suppose to have tetrachromacy. The article mentions "dinosaurs were endowed with the highly differentiated color vision of birds". Does this mean Dinosaur's "inherited" their tetrachromacy from birds ? 
No, birds are the closest living relatives to dinosaurs, and birds can be said to have inherited tetrachromacy from dinosaurs - see the cladogram below.





Why does the article mention reptiles before that ? Are birds reptiles ?
Birds are not reptiles, they are separate classes within the animal kingdom.
What is the direct connection between mammals and the dinosaurs ? ... why bring the mammals into the discussion ?
Here below is another cladogram; there is no obvious, direct (evolutionary) connection between dinosaurs and mammals. The thing is that dinosaurs ruled the world before the last mass extinction, thereafter it were the mammals. That's why the authors discuss them both, I guess. Especially since we are mammals it may make sense to draw parallels. 





What is the connection between pennaceous feather and planar feathers
As far as I could track the two down, they are the same (Prum, 2006)


Lastly, in response to your original question, before edits: Here is a link with a plain-English popular-scientific web article published online by the first author.