Sometimes when I tune a unison beat-less to the fundamental, I hear beats in the particles. Would you tune the partials beat-less or somewhere in between. Is there a way to get both the fundamental and particle of a unison beat-less? Is there something else that I am hearing?

Normally when there is a discrepancy like that you are tuning wound bass strings. This is due to flaws in the string/s or slightly different strings in the matched pair.
In this case the only way to resolve the problem is to either tune the beating partials that have the most volume beatless and or tune somewhere inbetween so the note is the least offensive.
Plain wire can also give similar problems if the wire is damaged, kinked or there is a bad termination.

Could grooved flat hammers be considered a significant source of "noise"?

With mis-matched bass strings needling the hammer can help. (If they are mis-matched and the problem isn't actually hammer to string mating.) The upper partials aren't as noticable then.

The piano that gives me the most trouble is a 1980s young chang, all original. I will check the hammers. I have to do regulation soon on it.

Our recent tuning class was on inharmonicity and we were talking about the very things Gene and Unright are referring to. Really fascinating stuff.