We think the following is true:
For all sufficiently large primes $p$ and all natural $g \ge 1$, there exists affine plane curve $f(x,y)=0$ over $\mathbb{F}_p$ which is non-singular, absolutely irreducible, of genus $g$ and it doesn't have any rational points.
Is it true?
Is it known?
This doesn't violate Hasse-Weil bound, because the bound requires SMOOTH projective model and our examples have very few singular points on the projective model.
For $p=13,g=1$ check this question and comments
Added Example of pointless non-sigular affine curves for $g=1$ defined by two equations. Let $f_0=x^3+y^2-1,f_1=z(x^p-x)-1$ and the curve $C : f_0=0,f_1=0$. Then $f_1$ is linear in $z$ so $C$ is birationally equivalent to $f_0=0$.
The curve is pointless because $x^p-x$ is zero modulo $p$.
To get a single equation for the curve set $f$ the resultant of $f_0$ and $f_1$ wrt $x$, experimentally it is irreducible.