What are your preferred profiles for integrated amplifiers

I would prefer a high end highly detailed, highly musical component like an Audio Research integrated amp. Incredible musical and detailed products... after I listened to it and read the reviews and bought it I would look to see if it had a fan, what class it was... etc. I guess the color you would want to choose first, but like all other aspects of the sound it is a matter of your tastes. So, I would concentrate on finding the kind of sound you want. You will find advocates and critics of all the aspects you brought up.
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Surprisingly to me, my Audio Research REF160s amp has a fan... I will admit I never thought I would have an amp with a fan.

1 - 2U or 3U
2 - External
3 - Black
4 - Class A

The one that sounds best. There are great products in each design class. 

Read the specs for a Balanced Audio Technology VK-3000SE. The perfect real-world Integrated Amp.
A bonus is the ease of bi-amping this design allows.

"The one that sounds best. There are great products in each design class."

+1

Class A, silver, external fins (artistically machined, a la Pathos' fins), 2u or 3u

What's 2u and 3u?

😂😂😂Better question: Who cares???!