Given a prime $p$ and degree $d$ is it possible to define polynomials $f(x,y)\in\mathbb Z[x,y]$ with total degree $d$ and number of roots at most $t$ where $t$ is any integer in $[0,B]$ for some upper bound $B$ that depends on $p$?
In particular can $t$ be $0$ or $1$?
Sure for $d\geq 2$. Just pick $f(x,y)=x^2-q$ where $q$ is a quadratic nonresidue mod $p$, for example, or if $p=2$ pick $f(x,y)=x^2+x+1$.
Note that an even more trivial example: $f(x,y)=1+pg(x,y)$ with $g(x,y)\in\mathbb{Z}[x,y]$ any degree $d$ polynomial, also give no zeros when reduced mod $p$. This is essentially the only example for $d=1$.