Installing washer/dryer combo in kitchen
No Way on the tailpiece. It's got to be a full glued on PVC Wye (that accepts 2") above the sink trap or with it's own trap under there. Definitely go with 2" (not 1-1/2" pipe), but the 6"-ish pitch is fine.

Fill your sink up half way &amp; drain it. If it squeals &amp; has a very tight vortex smaller than your finger &amp; the vortex starts &amp; stops, then you need more vent air for an already starved drain.

You're either too far from the vent pipe or that pipe is too small &amp; you'll need to put in an under-sink vent to avoid needing a utility sink that could hold the entire washer's dump that could then drain at a slower rate.