Were the medieval popes examined on a toilet-lid-like chair to establish their masculinity?

The chair was real, but its actual function is different from what the legend says. And actually, it seems to have been three chairs: one "commode"-type seat, and two porphyry "pierced" chairs. Read on for more.

In Misconceptions About the Middle Ages (page 66-7; ed. Harris and Grigsby), a description is given of the chairs' function in papal coronations:

The alleged manual examination of the papal claimant's genitalia, performed by a junior deacon through a [sic] opening in the papal chair, aimed, of course, to guard against the election of another woman Pope. ... By 1406, the humanist Jacopo d'Angelo ... denies in his report of the coronation of Gregory XII that such a rite existed, and accounts for the false belief along largely correct lines by explaining the meaning of function of various chairs in the coronation ritual: one chair at the Lateran, called the sedes stercoraria [which means "dung chair" or "toilet"], on which the Pope sits to remind him "that he rises out of clay and dung," that is, that he is subject to the failings of our human nature; and two porphyry seats in the Chapel of San Sylvester (which, because they are pierced, Jacopo says, "the common people tell the senseless fable that someone touches him as he sits to prove that he is indeed a man"). Seated in the latter, the Pope receives, respectively, the papal staff and keys (i.e., the power to govern, granted by Christ to Peter) and a red belt hung with twelve precious stones (invoking the Old Testament priesthood). The two porphyry chairs had first been noted in the investiture of Paschal II in 1099, as found in the Liber Pontificalis. ... The "stercory chair" was eliminated from the rite of investiture in 1560, while the two pierced chairs were last used in the accession of Leo IX [sic; that should be Leo X] in 1513. One was taken from the Vatican by Napoleon after the Treaty of Tolentino and is now in the Louvre; the other is still in the Vatican Museum.

The writing of Jacopo d'Angelo to which they refer is a letter to a certain Emmanuel Chrysoloras, which doesn't seem to be available in English. The editors claim indebtedness to The Myth of Pope Joan by Alain Boureau, so if you want to know more then that's probably a good place to look.

The renowned 19th century historian of Rome, Ferdinand Gregorovius, described the sedia stercoraria in further detail in History of the City of Rome in the Middle Ages, Volume 5, Issue 1:

Advancing across the forums and through the triumphal arches of Septimius Severus and Titus the procession skirted the Colosseum passed by S. Clemente and reached the piazza of the Lateran. Here the clergy of the Lateran received the pope with solemn song. They escorted him to the Portico where he took his seat on an ancient marble chair, the sella stercoraria. This symbolic ceremony of the deepest abas*m*nt of the supreme Head of Christendom on a seat bearing such a name is perhaps the most curious custom of the Middle Ages, a custom which we can only now contemplate with a smile. Cardinals however hastened to raise the Holy Father from the inappropriate seat with the comforting words of Scripture, "He taketh up the simple out of the dust and lifteth the poor out of the mire." [Psalm 113:7] The pope standing erect took three handfuls of gold, silver, and copper from the lap of one of his chamberlains and threw them among the people saying "Gold and silver are not mine but what I have that I give thee." [Acts 3:6]

In conclusion, though reports of the testicle verification rite go back a long way, so do contrary reports. The contrary reports make more sense (to me at least) and the reports of the testicular examination come chiefly from opponents of the Papacy and give confused accounts of what actually took place. My verdict, personally, is that it never took place but that someone saw the chairs with the holes and let their imagination do the rest.