- sexus (sex or gender)
The gist of Au101's answer is confirmed by de Vaan's Etymological Dictionary. First, regarding sex, in Proto-Italic and Proto-Indo-European, he gives:
PIt. *seks 'six', *seks-to- 'sixth'
PIE *(s)ueks 'six', *uks-ó- 'sixth'.
The PIt. form *seks has analogically dropped *-w- from *sweks by analogy with *septm 'seven'.
Regarding sexus, there is some question as to its origin. He writes:
Secus seems the more original formation, but it is strange that the older texts only know sexus. The modern meaning of sectiō 'division' suggests that sec/xus might derive from secāre 'to sever', but the morphology remains unclear: does sexus go back to an s-present *sek-s- 'to cut up', or was it derived from a form *sek-s- of the putative s-stem underlying secus?
He gives the Proto-Italic root of secāre as *sekaje/o-, and the PIE root as *sekhI-ie/o, meaning "to cut off". For secus, PIt. is *sekwos and PIE is *se-kuo, meaning "for/by oneself, separate."
So to summarize, even in PIE, the roots of these two Latin words were different words with distinct meanings – "six" in the case of sex and "separate" or "to cut off" in the case of sexus. So though they bear superficial similarities, the Latin words are etymologically unrelated.
No, I don't think so, and for this I can actually rely on etymonline which is a fine resource, even if linguistics students are discouraged from using it for their homework.
The entry for the English word 'six' is complete enough:
Old English siex, six, sex, from Proto-Germanic *sekhs (source also of Old Saxon and Danish seks, Old Norse, Swedish, and Old Frisian sex, Middle Dutch sesse, Dutch zes, Old High German sehs, German sechs, Gothic saihs), from PIE *s(w)eks (source also of Sanskrit ṣaṣ, Avestan kshvash, Persian shash, Greek hex, Latin sex, Old Church Slavonic sesti, Polish sześć, Russian shesti, Lithuanian szeszi, Old Irish se, Welsh chwech).
And then for 'sex':
late 14c., "males or females collectively," from Latin sexus "a sex, state of being either male or female, gender," of uncertain origin. "Commonly taken with seco as division or 'half' of the race" [Tucker], which would connect it to secare "to divide or cut" (see section (n.)).
And for secare we can actually cheat and indulge in use of the same resource. The entry for 'section' reads
late 14c., "intersection of two straight lines; division of a scale;" from Old French section or directly from Latin sectionem (nominative sectio) "a cutting, cutting off, division," noun of action from past participle stem of secare "to cut," from PIE root *sek- "to cut" (source also of Old Church Slavonic seko, sešti "to cut," se čivo "ax, hatchet;" Lithuanian isekti "to engrave, carve;" Albanian šate "mattock;" Old Saxon segasna, Old English sigðe "scythe;" Old English secg "sword," seax "knife, short sword;" Old Irish doescim "I cut;" Latin saxum "rock, stone")