In this paper, we settle a series of algorithmic problems related to the (sets of) subsequences occurring in the strings of a given formal language, represented by a deterministic finite automaton with translucent letters accepting it, which were left open in [Fazekas et al., ISAAC 2024]. Firstly, we show that one can decide in polynomial time whether all elements of a given language have a given string as subsequence. We continue by considering the problems of deciding for a given language \(L\subset \Sigma ^*\) and a given positive integer \(k>0\) (respectively, for all integers \(k>0\) ) whether there exists a k-universal string in L (i.e., a string which has all possible length-k strings over \(\Sigma \) as subsequences). For both problems we show that they are NP-hard, and in the case of the second problem we give an NP-upper bound and thus achieve NP-completeness. For the other problem, i.e. for the question whether there exists a k-universal word in L given a language \(L\subset \Sigma ^*\) and a positive integer \(k>0\) , we present a PSPACE algorithm.