  This article is twofold. In the first part of the article, we consider
Brezis-Oswald problem involving the mixed anisotropic and nonlocal $p$-Laplace
operator and establish existence, uniqueness, boundedness and strong maximum
principle. Further, for some mixed anisotropic and nonlocal $p$-Laplace type
equations, we obtain Sturmian comparison theorem, prove some comparsion
results, nonexistence results, establish weighted Hardy type inequality and
study a system of singular mixed anisotropic and nonlocal $p$-Laplace
equations. One of the key ingredient is the combination of Picone identity
already available in the separate local and nonlocal cases. In the second part
of the article, we are mainly concerned with regularity estimates. To this end,
in the elliptic setting, we establish weak Harnack inequality and
semicontinuity results. Further, we consider a class of doubly nonlinear mixed
anisotropic and nonlocal equations and prove semicontinuity results and
pointwise behavior of solutions. These are based on suitable energy estimates
and De Giorgi type lemmas and expansion of positivity. Finally, we establish
various energy estimates which may be of independent interest.
