Well, you're lumping a few things together that are worth thinking about separately.
If you want to just consider how stats affect the missions in the game, that's pretty cut and dry. The team you create for that mission (which might include your captain and first mate) would be the ones making progress, and it would certainly depend on their stats - in addition to facilities aboard the ship.
Right now, as a prototype, each mission has a number of points required, and a level of difficulty that divides the number of points you try to put into it. Missions frequently also have phases.
If you were going to scan BX9, you might need 30 points at 2 difficulty. Scanning missions require scanning hardware, such as the antennae I've already put into the game. Those antennae create 1 point per day on their own. If you had one antenna, it would take 60 days - each day you'd add 1 / difficulty to the total. At the end of that time, you would move into the next phase, which would probably require landing on the planet and doing more stuff like that down there.
However, that's with minimal attention. If you assign crewmembers to make the most out of the antenna, those specs can change. Right now it's set up so that the antenna will reduce the mission difficulty when manned, so manning it would cut the time down substantially depending on the relevant trait on the crewmember and how many stat-boosting items are near her bunkroom.
In the prototype, contested missions such as negotiations and tracking down enemy ships would have both sides run with the same stats, but inverted. That is, from my side the mission has points and difficulty, and from your side it has difficulty and points. The more progress I make, the higher your difficulty becomes, and visa-versa. Then, when either side reaches a specific number (100?) they win, and it's easy to tell how far ahead they were when determining the outcomes.
I haven't settled on this 100%, and things may change some, but you can see how it works in essence.