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 $a_1(\alpha )f_1(\alpha {)}^n+\dots +a_k(\alpha )f_k(\alpha {)}^n$ with the coefficients $a_i$ and characteristic roots $f_i$, $i=1,\dots ,k$, 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.