Warm, rich, smooth, 3d tube integrated amp, budget $6000

FYI tubes get super hot.  That's what they do.  The more tubes, the more heat.  If you want a "cool" tube amp, try solid state (but not Class A)!

I am a broken record on this, but, the integrated tube amp that I am a BIG fan of is the Synthesis Audio A-40 (40-watt, pushpull, KT-66 tube).  It comes with a killer good DAC included

I agree with the above statement. I have a Synthesis Audio A-40 that powers a pair of Audio Note speakers and the sound is just sublime. Feel free to PM me if you would like any additional info. 
john

Prima Luna Evo 400 fits all your requirements. And it drive pretty much any speakers. If you change the two front 12AU7 by NOS tubes you get even more sound space. Highly recommended.  

You should check out Trafomatic Audio amps. Very well designed and musically right sounding amps.

@fatdaddy2 I hear you... even my 9.2 watt elekit has some heat but much better than the nuclear fireball that is my pass x350.8. If it gets hot im ok with it... i dont think ill get the sound I want with a cool running amp.