Do problems that have unary encodings automatically become unary languages?
A unary language is a subset of $\sigma^*$ for some letter $\sigma$. If your language is of this form, then it is unary. Otherwise it isn't.

A unary encoding is an encoding of some data as a subset of $\sigma^*$. For example, one possible unary encoding of the natural numbers is $n \mapsto 1^n$. Any language in which instances are unary encoded is a unary language, simply because it is a subset of $\sigma^*$. For example, the language of all primes encoded in unary is a unary language.