Tube over tube.... Could this combination becomes.....over tubey ???

If youve counter balanced with the right cables and headphone selection, then no not at all. But if you go with too smooth sounding copper cables, and a dark headphone like a Denon or JVC, then yes, to some ears this would become like maple syrup, delicious, but only on pancakes, and not much else.

Let us know what else you are using for cables and what your headphone of choice is.

That really depends on the gear you choose. The stereotypical tube sound is mostly what you get from triodes on output transformers.

I un all tubes now and then - usually the turntable through the Yph phono stage into the Zana. The Zana doesn't sound tubey, but the Yph adds a little character from its tubes. On the other hand, my Scott 350B tubed stereo FM tuner doesn't add much character. That fed into the Zana isn't terribly noticeable.

So find out what each piece of gear sounds like and you'll know.

I've never noticed a cable to have any bearing on how a circuit in a discrete component works. If a tube amp makes the low end flabby, you can actually see and measure that. You'll still get the same measurement of a flabby low end no matter the cable. I suppose you could argue that something else is going on, but you'll still be left with the same frequency measurement demonstrating flab.

With the HD 800s as your can of choice, then I would bet the overall sound would be pretty well balanced.  From what I've been able to make of the 800 the few times I've been able to listen to it, its lit up or ever so lightly bright treble, and its somewhat absent lower bass would benefit well from your proposed rig.  I would say go for it! 

Have you read Srajans review of the EE dac on 6moons? If not I would suggest you read it, if only to get a better idea of what it is. He too address the "tube" sound of the unit.