I thought' I'd answer the original question before I read anyone else's comments (sort of voting blind), then look for common ideas.
1. Overall, what is the ship like? Is it old and unsafe? New with hidden problems? Cobbled together from scavenged parts? Big and cavernous? Small and cramped?
I like a ship with quirks because it means that if anybody wants to play a mechanic or engineer, there's plenty to do. It also means that your home base is tricky and presents surprises; on the other hand, most of the time it shouldn't be a death trap. Serenity, the TARDIS, and the Millenium Falcon all gave us examples of somewhat recalcitrant vintage vehicles that will outperform others in the same class if they are treated with love and care; by contrast, the Entreprise in all versions was always a shiny marvel that was tricky precisely because it was at the bleeding edge of technology.
I lean towards older and temperemental with lots of retrofits. I definitely like a ship with lots of nooks and crannies for surprises, stowaways, contraband, etc.
2. What is the major problem with the ship? Slow and lumbering? Temperamental hyperdrive? Annoying computer? Cargo bay doors occasionally malfunction?
I was thinking fast under full power, but slow to accelerate and decelerate, making it an important decision to commit to a course. Inconveniently-configured cargo space is also interesting, making transport of large quantites of material or personnel an excercise in space-Tetris.
3. What is the ship's hidden strength? Nooks and crannies for contraband? Able to withstand punishment? Oversized power plant? Hidden weapons pods? Sporty paint job?
I've already spoken in favour of nooks and crannies, and high acceleration under full power. I also like the idea of a surprisingly advanced sensor suite, which would cause the ship to all sorts of missions where we have to listen quietly to suspicious chatter or perform deep-scans of odd alien technology remnants.
4. What sort of person is the Captain? Male, female, or other? Why did they end up commanding a Class D?
I'd like a captain who either (A) hates our guts, or (B) has an overactive sense what Bulldogs' duty entails, because in either case it will mean placing us in a lot of dangerous situations. I'll vote for a stick-in-the-mud by-the-book career officer, a Templar maybe, who has just arrived on board recently and replaced our beloved easy-going former captain after she was killed. The new captain is out to shake this crew of slackers and turn us into military personnel or cannon fodder. Everybody clean up their uniforms!
5. What is the Captain's Trouble? Secret past? Bad marriage? In debt to the mob? Terminal disease?
The captain has mysterious trouble with Templari religious authorities; they refer to her/him as a heretic.
6. What is the Captain's command style? Draconian? Military? Lackadaisical? Disorganized? Professional?
Everything ship-shape and by-the-book — at least, that's what it looks like to us slackers. But in a bar on Central Point, I heard a group of mid-level Templar Navy officers call him/her a slob…