On the Skolem problem and some related questions for parametric families of linear recurrence sequences

We show that in a parametric family of linear recurrence sequences $a_1(\alpha) f_1(\alpha)^n + \ldots + a_k(\alpha) f_k(\alpha)^n$ with the coefficients $a_i$ and characteristic roots $f_i$, $i=1, \ldots,k$, given by rational functions over some number field, for all but a set of $\alpha$ of bounded height in the algebraic closure of $\mathbb Q$, the Skolem problem is solvable, and the existence of a zero in such a sequence can be effectively decided. We also discuss several related questions.