Does a stressed ship that dock retain its stress while in reserve? i.e. If my YV-666 with Hound's Tooth title docks a stressed Binayre Pirate, will that Pirate still be stressed if I deply it again two turns later?

I don't see anything in the rules that would suggest the stress should go away (Reserved rules indicate that circular tokens go away as normal, which implies that tokens continue to exist on a ship when docked) so I assume the answer is "yes," but it seems odd to me that a stressed pilot is sitting in the docking bay quaking in fear instead of mentally regrouping and maybe checking on his fantasy podracing league standings.

"During the End Phase, a ship that is in reserve still removes all circular
tokens and recovers charges on all of its cards with recurring charge
icons." - page 16 of the rules reference.

Now this has nothing to do with removing tokens, but since there is nothing stated about removing the stress I would say that it keeps it while in reserve. The above-quoted section shows tokens are still on things and interact with the rules while in reserve, therefore it would make sense that the stress stays on there.

Docked ships keep all their tokens and whatnot.

Being in reserve has very specific effects:

Because you don't get a dial, and don't do maneuvers, you can't remove stress via blue maneuvers. I believe there is no other way to do so either, while the ship remains in reserve.

So, yup, pilot stays freaked out the whole time. Similarly, they are unable to put out console fires or patch hull breaches. Not sure what they're doing in that docking bay.