There's nothing in the episode to say it didn't have warp capability.
And what would be the point in stealing a vehicle without warp drive?
Considering how long-lived the two Cheronian clowns were, a sublight trip from star to star would be a perfectly viable maneuver for them, even if it took a few thousand years...
Was it 200 years ago? Nearly 200 years ago?
Kirk in the teaser says the ship has been missing for "over two centuries". Later he says the recorder marker they recovered was launched "two hundred years ago". That's the extent of it - but it doesn't seem to support the 170 year interpretation, as the "over" part is quite explicit. Probably we should be thinking in terms of 202 or 207 years or something like that...
On the issue of the Enterprise
being within easy reach of Delta Vega after the misadventure, it could be chalked up to pure coincidence. Perhaps the galaxy is full of such automated outposts (not necessarily of Federation origin), especially at the rim, in support of an aggressive expansion program? Perhaps Kirk actually set out using Delta Vega as his "last call saloon", one of the many supply outposts that his ambitious expedition required?
Speculating further on that vein, Vega is a real star but nowhere near the outer regions of the Milky Way. Also, Vega is not a constellation, so Delta Vega should not denote the fourth brightest star in the constellation Vega. Perhaps it's just a codename? Kirk's expedition might have relied on the robotic outposts Alpha Vega, Beta Vega, Gamma Vega, Delta Vega and Epsilon Vega, all clustered near his "exit point" - and the barrier phenomenon might have spat him out within a few lightdays of the fourth of those, rather than within a few lightmonths of the second or a few lightminutes of the fifth.
If Delta Vega is a code name (or a trade name, like Dytallix B of TNG fame), then the Delta Vega outpost of STXI becomes all the more acceptable: neither the TOS nor the STXI installation is in any way related to the star Vega, and it just so happens that the Starfleet of TOS assigned that code name to a different outpost from the Starfleet of STXI.