  Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite
surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and
Adelic Zariski Dense Orbit Conjecture (AZO). We consider also
Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical
degree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ at a point
$P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine
surface, such that the log Kodaira dimension $\bar{\kappa}(X)$ is non-negative
(resp. the \'etale fundamental group $\pi_1^{\text{\'et}}(X)$ is infinite), we
confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp.
AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every
surjective endomorphism on any projective variety with ''larger'' first
dynamical degree (resp. every dominant endomorphism of any semiabelian
variety).
