Can it down-feed 0.001" or less?
Yes - with a couple caveats.
1.)put an indicator on it.
Either the toolslide is left relatively loose and there is enough mass that it hangs on the lead (downfeed) screw. Lock it securely before commencing the cut.
Or with a snug slide/gibs you have to feed down with the leadscrew...oops!.... and raise back up to take the slop out.
If it is a light cut, that is probably adequate.
If a heavy cut, maybe bring the leadscrew back to bear again, and "estimate" how much pressure to add to keep the toolslide in place.
Cuts on a shaper are subject to the normal way wear any machine is. It was traditional to make cuts for parallel features on work done on a shaper by swapping the work end for end to make both sides of the feature with the same tool set up. This saves time for tool changes (LH to RH, e.g.). But it also avoided the factor that with side pressure, the advancing ram deflects opposite directions if the cut is made first on say the LH side, then the opposite hand tool installed and the RH side cut on the opposite side of the ram. Shapers are/can be quite good for accuracy, but some of the geometry has to be thought through "differently". (FWIW, i probably usually accept the tool change (or use a duplex tool) & do a parallel feature from both sides rather than disturbing a vise set up. The taper is usually quite small if it exists. Certainly on short parts)
Then there is the added factor of the clapper.
Caveat 2.) if the clapper is worn & floating in its box or especially if the pin is loose in any way*, the clapper will rise or diverge in the cut and be un-predictable.
Lifters save wear on the clapper and tool because the tool does not drag on the backstroke, and the clapper does not bounce 7 or a dozen times on the way back either. But they also allow running the clapper more snugly for precise cuts.
smt
*both the taper pin and the hole wear oval, and the hole can go bell mouthed.
So on a worn clapper, it might not be possible to tighten the pin enough to avoid divergence in a cut and still have the clapper function under gravity alone. Besides the fact that the ears of the box probably need reamed to get the pin to actually seat deeper anyway.