A, The ship will be smaller than the Walrus and designed more for speed and maneuverability, especially since Tevinter is used to having to evade Qunari dreadnoughts to avoid engaging with their gaatlok cannons. What type of ship specifically is a bit beyond us (and Thedas naval technology is on a different historical path than our world's), but you'd be looking at something in the size range of the ship types on the first row of this image and designed for a maximum crew of about 40.

B, The ship would remain Inquisition property! Who was in charge of its day to day stewardship/default captaincy would be up to Araceli, and if the person generally captaining it for the Inquisition were unavailable for a mission that required use of it then the ship would be used without them if necessary.