For staff ignore the talent, experience and morale values because they don't do anything, it's all about the star ratings.
CD: You really need 4.5 or 5 stars with a good balance of strength and weakness. The CD influences design (and therefore indirectly speed) of your cars at the beginning of next season. The CD also modifies research power. If you carry out research in the CD strength the power will be higher than indicated, the opposite happens if you research in your CD weakness. For a more detailed explanation see here...
Next season's car
TD: Influences your base research power. Again employing a TD with more stars will increase your base research power. This is also influenced by CD as described above.
Doctor: Influences health loss during training. High star rating means that river health decreases less with each press of the training button. IMO the difference is not significant enough to justify the high wages of a 4.5 or 5 star CD. Personally I just let the Doctor contract expire and the game auto assigns a new doctor with a wage of £50k per race. Over the entire season this will save you in the region of £10m to £12m.
The star rating is based on a comparison between your level as a manager and the level of the staff member. If you are L14 and the staff member is also L14 they will have a 4.5 or 5 star rating. Because staff do not level up, every time you level up as a manager the star rating of the staff members will reduce. This is something to consider very carefully when renewing contracts of older staff, especially if you are new to the game and levelling up quickly. Worse case scenario is you renew a contract of an aging CD at which point they announce their retirement at the end of the 50 race contract. You're now in trouble, their star rating will gradually decrease as you level up but to get rid of them is going to cost you a small fortune.
As far as I know, other than improving the car the staff have no influence on the race. That is to say they don't help with qualifying, car pace or pit stop time.
I hope this helps
Kev.