Recommend A New DAC Between $1000 and $2000

Denafrips PontusII was very good for me. 

For your budget and checklist: Denafrips Pontus II near the top of the range and Ares II just over the bottom of the range. Warm rich and will show you why R2R DACS rule.

I'd recommend a RME ADI-2 FS as a balanced DAC with a great remote.  This DAC preamp is unique in its features and ability to move the sound shape and to do parametric EQ digitally upstream of the analog signal.  You can convert your headphones or speakers to basically anything you would like it to be.  The remote has very fine incremental L/R Balance, Bass and Treble control.  You also can control the parametric EQ individually on the Left or Right Channel - great for hearing loss in one ear or EQing individual speakers in a room to compensate for room placement.  Or just use it like a great DAC on pure line output or a high power single-ended or very low noise IEM DAC/AMP combo.  Providing a DAC with variable volume control (with any EQ)  to a pair of powered speakers, it is impossible to beat in my opinion.

I second the Rme adi2 suggestion. It is a gem.