I have a Bottlehead Crack with Speedball and some other mods, and I use it with a HD600 (also AKG K501). I don't find it excessively warm at all. Re your first question about a suitable amp for the HD600, it really depends on the sound you're looking for. Personally, I find the HD600 have a rather "closed-in" sound, especially compared to my AKGs. I do like their mid-range, I just wish it the sound stage was wider. Having said that, I think the Bottlehead pairs up quite well with the HD600, and I do prefer that amp over the Millett MiniMax and the iBasso D10.
To answer your second question about tubes... By "lively", I assume you mean like a more forward presentation, more analytical sound? In terms of signal tubes, I think the 12BH7's have a livelier presentation than the 12AU7's (although the Telefunken 12AU7 is really quite analytical). For power tubes, 7268 > 6080 > 6AS7G.