I would vote for Hegel H360 as well. However, one thing to note is that there is nothing warm about the Hegel. It's all about high resolution, detail and clarity. It does have very good punch.
I think it would be the same for both the Lyngdorf and Moonaudio units, as they are also about super high resolution and clarity. But I still think Hegel would be superior here.
The only item I know of that meets your requirements is the Pass Labs INT 250, but that is way above your budget ($8500 used), and it does not include a DAC. But it checks all your sound quality items.
Under $5000, the only other thing I can think of is the Parasound Halo or new INT6 integrated amps, but I recommend you load it with a Furutech or Hi-Fi Tuning Supreme fuse. At a retail of $3k, it's well under your budget, but it's probably not to the quality you want. The preamp and DAC is not that great either.
Actually, if you go with Focal speakers, I would totally avoid Lyngdorf and Moon Audio as they will likely sound way too thin. The Hegel may sound thin as well. With Focal, I would recommend the Pass Labs or Parasound type of amp.