Description for languages that can be solved in time(n)? How can one describe all languages that are in
$\mathrm{TIME}(n)$? 

It can't be all the regular languages only, as for example $L = \{a^n b^nw \mid w \in \Sigma^* \land n \geq 1\}$ is not regular but in $\mathrm{TIME}(n)$. Furthermore there are regular languages (like any modulo languages) that are not in $\mathrm{TIME}(n)$. 

So, is there a nice description for all $L \in \mathrm{TIME}(n)$?
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).