As for Time Agents, we still never got any clear answers on what exactly the agency was, what its function was, etc. Jack's vortex manipulator shorted out rather easily. Their time travel capabilities might have been laughable at best.
To be fair, Jack was from the 51st century, so, depending on the Time Agency's jurisdiction, it might've been rated to only travel a few thousand years in a hop. Going from the 201st century to the 21st (or trying to) was probably a couple of times beyond it's design capacity (which would also explain the hundred-fifty year margin of error).
Still, the machine itself was obviously more than sound, since the Doctor only need a couple minutes of jiggery-pokery to get it to go 100 trillion years into the past, to a specific city on a specific planet, probably an ungodly distance away. It's also likely that the Time Agency had capsules (not full-on TARDISes but some kind of time-ship), and they may have had more robust non-portable vortex-manipulators that they normally used for missions, the wrist-strap being more of a back-up than a primary mode of travel.
My guess is that the Time Agency were, well, cops. Sort of like the Time Lords were, sealing breaches, preventing history-altering disasters, that sort of thing, but with a more local purview. They knew
of the Time War, and remembered first-hand stuff with the Daleks (and Jack's implied that he's met Time Lords other than the Doctor before, or at least read a couple books on them), so they were probably middle-to-upper tier in terms of their understanding of time-travel and the Causal Nexus (lower-tier having not noticed anything at all of the Time War and being completely oblivious to how the Daleks could be extinct in 2005, returning in 2008, extinct again in 2012, and at the height of their empire in 216, middle-tier having heard of the Time-War from other people but not believing it, and upper-tier having a full conception of the timey-wimey ball, and direct knowledge of or participation in the Time War).
Speaking of, we can add the Chula to the list of people who had time-travel, and presumably whomever they were having a war with.