Can we say that "he" and "she" are cognates?

You asked two three questions.

1. Are he and she cognate? The text you copied from etymonline says that he derives from Old English he, which derives originally from a reconstructed Proto-Indo-European root *ki—, whereas she derives from the feminine form sio of Old English se meaning "the" or "that", which according to the OED derives from a reconstructed P.I.E. root *so–. So these words are not cognate, at least according to etymonline and the OED.

2. How did the sound change from sio to she happen? The OED explains it like this:

   It would appear that in some dialects of late Old English the diphthong in this word underwent a change of stress, the older pronunciations /siːo/ [nominative] and /siːe/ [accusative] being replaced by /sjoː/ and /sjeː/.... As the combination /sj/ is acoustically close to /ʃ/ , and more difficult (according to English habits of articulation) to produce, it is not surprising that /sjeː/ /sjoː/ became /ʃeː/ /ʃoː/.

   And then, of course, the Great Vowel Shift changed /ʃeː/ to /ʃiː/.

3. What can we make of the disagreement between Wiktionary and the OED? Well, there are clearly two competing theories for the etymology of she.

   In the OED's preferred theory we have she < O.E. sio (feminine form of se meaning "the" or "that"). The OED suggests that the displacement of heo by sio took place in the north of England, and in support of this it cites the "Lindisfarne gospels and the glosses to the Durham Ritual and Hymnarium".

   In Wiktionary's theory we have she < O.E. heo (feminine form of he meaning "he"). The OED discusses this theory in addition to its own preferred one:

   Some scholars have maintained that she and its dialectal variants descend directly from the pronunciations /hjeː/ /hjoː/ of heo; the contention being that /hj/ might naturally develop into /ʃ/. This development has occurred in some Norwegian dialects, and it is illustrated by the proper names Shetland and Shapinshay from Old Norse Hjaltland and Hjalpandisøy. There is slight support for this view in the existence of northern dialect forms such as shoop representing Old English heope. Other views are that /ʃ/ was substituted for the un-English sound /ç/ , developed from /hj/ , and that it arose from the sequence –s + j– in such contexts as was was hió.

   So there's evidence for both theories, but not enough for one to be overwhelmingly preferred to the other (or perhaps both theories are partly true, or describe the development in different places). You may have to be prepared to accept that the answer is not known with certainty.