Re: The Department of Temporal Investigations.
As for what it does, not much is known about that since the novels are not canon and hence none of any of their content is applicable, and it received minimal screen-time.
Why can't we include non-canonical information? Canonical information can be contradicted by new canonical information, too (James R. Kirk to James T. Kirk, the lack of any reference to the NX-01 in previous Trek productions, etc.), so who is that any different from non-canonical info?
In my opinion it were Voyager and later Enterprise (not Deep Space Nine) who, with their invention of the Temporal Integrity Commission, took the real life implications of time travel way too seriously. If time travel really was possible, I very much doubt there would be any such agency.
Are you kidding me? Of course
there would be an agency to try to monitor and control instances of time travel. That sort of technology would be incredibly
dangerous. Hell, you can't drive a car without dealing with a government agency trying to regulate your driving -- you think they'd let people be able to change history without trying to control that sort of technology?
"Every line of serious work that I have written since 1936 has been written, directly or indirectly, against totalitarianism and for democratic Socialism, as I understand it." - George Orwell, 1946