It's not so simple that you can generalize that one kind of amp topology will sound in a particular way, though in the most profoundly over-generalized non-contextual view, one could say that tube amps are warmer and take the edge off, while solid state is more clinical and crisp. Sorta. Not really. No. It depends. Whatever.

The sweet spot is honestly what your budget allows for that doesn't hurt your necessary/discretionary spending. For me, I try not to spend more than a mortgage payment on any piece of gear. This whole hobby is a horrible horrible game of diminishing returns. Spend only as much as you won't miss.

A note on the integrated amps: Most have headphone outputs that are tapped off the speaker outputs with a resistor in line. This will generally have an impedance interaction with the headphone, and in particular the Beyers have a wild impedance spike in the midbass so you will wind up with a boost in the frequency response in that region. An option is to hook up your headphones directly to the speaker posts with an adapter, but this might require some recabling of the headphone (depends on the amp).

I'll let someone else chime in on the specific amp pairings, as I haven't heard it with any of the specific amps in your list. I do recall liking the DT880/600 on a Bottlehead Crack w/Speedball though.

Yea, you can't go wrong with Valhalla 2 with any high impedance headphone.

I've heard that the valhala is bright and not very tubey.

I'm guessing that the valhala isn't 'neutral'. Can you describe what the valhala does to the sound?

How does it compare to the little dot series and the crack?

How do solid state amps compare?

I think it's actually quite neutral, with a tasteful bit of tube wetness, warmth & a nice sense of body with high impedance cans - tubes done well.  I wouldn't call it bright, but it's not as "tubey" as the Crack (in the HF sense of the word), nor as rolled in the treble. The Crack to my ear is actually the less neutral of the two. Vali is brighter IMO, if that provides another reference point.   

I'm looking for something that'll slightly tame the treble and make the whole sound more like the sennheiser hd600.

I dislike but don't hate DT880. It has good bass for a dynamic and adequate soundstage; it's also comfortable. But it's still somewhat bright even with tubes. Having heard Val 2 with other phones, I don't think it'll do what you want. I've heard it with Crack and I'm fairly confident it will do a better job, as would a Darkvoice. But if that's your goal, why not simply replace it with HD600?

But if that's your goal, why not simply replace it with HD600?

This man, he speaks truth.