Yes, rating stars have an effect on next seasons car, but next seasons car starts to be build at the first race of the active season so changing CDs midseason will only have a partial effect depending at how many races he/she was in charge.
Blunion said all we know but admittedly your CD does looks off. With Talent 19, Experience 16 and Morale 20 he should have leveled if staff levels at all. Level 13 for that guy is way too low unless those stats are just random numbers without effect and level is just a value determined once at start of the career and then fixed. Would be a shame, and misleading because why are they there at all then, if it really is done that way.