  We report the properties of more than 800 bursts detected from the repeating
fast radio burst (FRB) source FRB 20201124A with the Five-hundred-meter
Aperture Spherical radio telescope (FAST) during an extremely active episode on
UTC September 25th-28th, 2021 in a series of four papers. In this fourth paper
of the series, we present a systematic search of the spin period and linear
acceleration of the source object from both 996 individual pulse peaks and the
dedispersed time series. No credible spin period was found from this data set.
We rule out the presence of significant periodicity in the range between 1 ms
to 100 s with a pulse duty cycle $< 0.49\pm0.08$ (when the profile is defined
by a von-Mises function, not a boxcar function) and linear acceleration up to
$300$ m s$^{-2}$ in each of the four one-hour observing sessions, and up to
$0.6$ m s$^{-2}$ in all 4 days. These searches contest theoretical scenarios
involving a 1 ms to 100 s isolated magnetar/pulsar with surface magnetic field
$<10^{15}$ G and a small duty cycle (such as in a polar-cap emission mode) or a
pulsar with a companion star or black hole up to 100 M$_{\rm \odot}$ and
$P_b>10$ hours. We also perform a periodicity search of the fine structures and
identify 53 unrelated millisecond-timescale "periods" in multi-components with
the highest significance of 3.9 $\sigma$. The "periods" recovered from the fine
structures are neither consistent nor harmonically related. Thus they are not
likely to come from a spin period. We caution against claiming spin periodicity
with significance below $\sim$ 4 $\sigma$ with multi-components from one-off
FRBs. We discuss the implications of our results and the possible connections
between FRB multi-components and pulsar micro-structures.
