Furthermore, what happened to Continental ships captured by the Royal Navy? It's mentioned that the USS Providence(as an example) was captured and sold by the British, but what happened after it was sold? Who bought it and what was its fate?
They did not, the Continental Navy was in the end a pretty terrible fighting force.
Part of it was just being up against a competent foe and terrible starting point, some of it was poor management and lack of support from Congress.
Upon being founded one of the largest problems was the lack of purpose built warships. Congress ordered 13 frigates of various gun ratings, and 8 of them did make it to sea in some form.
But on the whole they were relatively poorly performing. CN crews and skippers were not wholly experienced and there was little shore establishment ready to support the effort. While gaggles of merchantmen were captured and few light vessels, no Royal Navy ships of 3rd Rate or larger were lost to American ships during the war.
The closest that one came was when HMS Yarmouth, a 3rd Rate that had some of her guns removed and escorting a convoy, was engaged by the USS Randolph. During the battle, despite suffering some damage, Yarmouth managed a lucky hit and Randolph's magazine went up, destroying the ship and most of her crew.
The CN also had success using converted merchant ships. While less stoutly built and obviously not designed with larger armaments in mind, with the right refits and a good captain they could still defeat RN frigates. John Paul Jones' Bonhomme Richard is a perfect example here.