Does the type of captain you pick (tac, science, eng) impact your starting ship? Science = science vessel, etc?
No. Your starting ship is always the same. Once you hit Lt.Com you can choose between the three tiers. Again, no "class" restriction. However, I would say that you get more of a boost to certain abilities by using the ships that match with your captain. A science officer will get more out of a science ship, for example, then a tac officer. But that doesn't always mean it's better, just depends on you really.
And some combinations work better than others, depending on play style. A Tactical Captain in a Cruiser can do very well with beams, since he can pack (and power) so many of them. An Engineer in a Science ship can be a huge asset to a team, as Science boffs get all the best Shield heals and Engineers get the Hull heals.
A Science Captain in an Escort might be a stretch, but I'm sure somebody now is going to reply that they had great success with this combination.
