  We present estimates of line-of-sight distortion fields derived from the 95
GHz and 150 GHz data taken by BICEP2, BICEP3, and Keck Array up to the 2018
observing season, leading to cosmological constraints and a study of
instrumental and astrophysical systematics. Cosmological constraints are
derived from three of the distortion fields concerning gravitational lensing
from large-scale structure, polarization rotation from magnetic fields or an
axion-like field, and the screening effect of patchy reionization. We measure
an amplitude of the lensing power spectrum $A_L^{\phi\phi}=0.95 \pm 0.20$. We
constrain polarization rotation, expressed as the coupling constant of a
Chern-Simons electromagnetic term $g_{a\gamma} \leq 2.6 \times 10^{-2}/H_I$,
where $H_I$ is the inflationary Hubble parameter, and an amplitude of
primordial magnetic fields smoothed over 1 Mpc $B_{1\text{Mpc}} \leq 6.6
\;\text{nG}$ at 95 GHz. We constrain the root mean square of optical-depth
fluctuations in a simple "crinkly surface" model of patchy reionization,
finding $A^\tau<0.19$ ($2\sigma$) for the coherence scale of $L_c=100$. We show
that all of the distortion fields of the 95 GHz and 150 GHz polarization maps
are consistent with simulations including lensed-$\Lambda$CDM, dust, and noise,
with no evidence for instrumental systematics. In some cases, the EB and TB
quadratic estimators presented here are more sensitive than our previous
map-based null tests at identifying and rejecting spurious B-modes that might
arise from instrumental effects. Finally, we verify that the standard
deprojection filtering in the BICEP/Keck data processing is effective at
removing temperature to polarization leakage.
