The trip from Deep Space Nine to Trill was 37 hours from Deep Space Nine in the Defiant (DS9 "Equilibrium", Season 3 episode 4). The mortally-wounded Jadzia Dax could not survive the trip (which is why Ezri became the Dax host - she was the only Trill aboard the USS Destiny, the ship assigned to take Jadzia from the station to Trill - DS9 "Shadows and Symbols", Season 7 episode 2).
As for if the vehicle made a difference, yes. Runabouts may be Warp-capable, but most Federation starships can outpace a Runabout at warp. This is because a starship can generate more power, and put that extra power into significantly larger warp nacelles, allowing for faster flight. Runabouts are typically only preferred because their size makes them harder to detect and (if attacked) be hit by enemy weapons.
Well I always thought Jadzia was actually dead by this point and that they just transported the symbiont back to Trill. According to Ezri "the Dax symbiont took a turn for the worse" sounds like just the symbiont was aboard which would be possible because a generally a symbiont or its Trill host only are in mortal danger if they are seperated for a couple days (or at least many hours longer that the travel from the station to Trill).
In terms of the speed of Runabouts and larger vessels one has to remember that Runabouts were designed to bridge the gap between shuttle craft and mid sized starships. So naturally they have a much shorter range and maximum speed because their warp and impulse engines are smaller. While this smaller size does give them greater stealth potential I think Starfleet just uses them for short range transport and science missions not necessarily combat. Think of the difference between a modern day lifeboat(shuttle), a small yacht(runabout) and a destroyer(a larger ship ex:Akira class).
However, if it were not a dire emergency, the ship would be limited to warp 5, which is also the maximum speed of a runabout