Description for languages that can be solved in time(n)?
We're not good at characterizing languages that can be decided in deterministic polynomial time. There are multiple characterizations of $\mathrm{NP}$ (e.g., nondeterministic Turing machines, polynomial witnesses, existential second-order logic), but $\mathrm{P}$ seems to be just "stuff you can do in polynomial time on a Turing machine."

Similarly, it's unlikely that there's any alternative characterization of linear time beyond its literal definition.


  Furthermore there are regular languages (like any modulo languages) that are not in $\mathrm{TIME}(n)$.


No there aren't. You can decide any regular language in linear time using a DFA (trivially modify its description so it's a Turing machine).