Wow. I would love to see more of the Hawking.
I'm a little confused about how to place what you're doing here. This is some sort of crossover between the STAR TREK "prime Universe" and some kind of Dalek-centric "alternate reality" of the DOCTOR WHO Universe, right? I'm also confused about "when" this story/stories would be set. So, Cassie is a Federation officer stationed on a starship, or a starship captain, or some kind of Federation agent?
Part of me thinks Cassie belongs on some kind of scout-class vessel, about the size of Forbin's Minmus or something smaller. Of course, depending on the era and the position Cassie holds, I could see her on a ship like Sisko's Defiant.
To answer your questions in that order:
The story is a cross between the Universe of Star Trek (pre-JJ Star Trek, least up to when Romulas snuffs it), and Doctor Who. What I mean by that is you'll see Daleks, you'll see Cybermen, and maybe the Doctor himself.
The time it's in is a decade after Nemesis, and just when Romulas is gone and Spock's apparent demise.
Cassy is of a species she knows nothing of, she was raised by humans on a colony world in the Neutral Zone. It was established by one of those expeditions that was forgotten, believed to have perished. Something like those in "Up the Long Ladder", "The Ensigns of Command", "Masterpiece Society" and so on. This colony, though is far from Federation ideals, in that remnants of the the power mongers (Cybus industry from our universe runs it like 'The Company' from Aliens or OCP from Robocop...where we see what Earth would have been like had not the Vulcans came, even some Terra Prime folks founded the government) run it. She has no clue as to what species she came from, due to having been found in a basket one night on that planet, with a note to take care of her.
She is totally brilliant, and has traveled with the Doctor, several times, and she is not really part of Starfleet, but helps out, like how the Doctor helped out with UNIT.
She eventually made a small, custom made ship using technology she learned from the Doctor, the Daleks, and some of her own ideals. She's sorta like Wheeljack from the original Transformers, Gadget from Rescue Rangers, The Doctor, and Washu from Tenchi. And she'll use that ship to arrive at the Starbase that the main hero ship is going to be at. And that ship is able to be run by just herself and her robot sidekick.
And as for the hero ship, it's going to be one made for deep space exploration....with no more Neutral Zone, I can see the exploratory programs going up more. And Laguna, the Vulcan captain (not raised by Vulcans), is going on her first command.