Is the fast of one who swallowed something from the skin of his lips with his saliva, or licked lips that had the taste of toothpaste on them after Fajr, valid? And is it permissible to count the extra days he fasted due to waswasah (obsessive whispering) as qada' (make-up) for past Ramadan days?

You are afflicted with Waswas (obsessive doubts). The cure for it is to turn away from it and ignore it. Swallowing saliva does not break the fast. Swallowing peeled skin from the lips unintentionally does not invalidate the fast unless it is swallowed deliberately. Fasting with the intention of making up for Ramadan is sufficient, whether it is for one Ramadan or multiple Ramadans, and there is no need to specify the reason for making up.