The first two are based on wiktionary
From Middle English false, from Old English fals (“false, fraud, falsehood”), from Latin falsus (“counterfeit, false; falsehood”), perfect passive participle of fallō (“deceive”).
Uncommon before the 12 century, the word was reinforced in Middle English by Norman fals (compare Old French faus), eventually displacing native Middle English les, lese (“false”), from Old English lēas; See lease, leasing.
For spelling, the -e (on -lse) is so the end is pronounced /ls/, rather than /lz/ as in falls, and does not change the vowel (‘a’). Compare else, pulse, convulse.From Proto-Indo-European *gʰwel- (“to lie, deceive”). Cognate with Ancient Greek φῆλος (phẽlos, “deceitful”), Sanskrit वृ (vṛ, “twist, crook”), Avestan 𐬰𐬎𐬭𐬀𐬵 (zurah, “injustice”), Lithuanian ẑulas (“rough”), Latvian zvel'u (“to turn aside”), Old Church Slavonic зълъ (zŭlŭ, “evil”)
But here is another etymology from myEtymology.com
the English word false
derived from the Latin word falsus (wrong, lying, fictitious)
derived from the Latin word fallere (deceive; slip by; disappoint)
derived from the Latin word facere (to make; act, take action, be active; compose, write; classify; do, make; create; make, build, construct; produce; produce by growth; bring forth)
derived from the Proto-Indo-European root *dhē-
The third etymology is from etymonline.
late 12c., from O.Fr. fals, faus (12c., Mod.Fr. faux) "false, fake, incorrect, mistaken, treacherous, deceitful," from L. falsus "deceived, erroneous, mistaken," pp. of fallere "deceive, disappoint," of uncertain origin (see fail). Adopted into other Germanic languages (cf. Ger. falsch, Du. valsch, Dan. falsk), though English is the only one in which the active sense of "deceitful" (a secondary sense in Latin) has predominated. False alarm recorded from 1570s. Related: Falsely; falseness.
early 13c., from O.Fr. falir (11c., Mod.Fr. faillir) "be lacking, miss, not succeed," from V.L. *fallire, from L. fallere "to trip, cause to fall;" figuratively "to deceive, trick, dupe, cheat, elude; fail, be lacking or defective."
Related: Failed; failing. Replaced O.E. abreoðan. The noun (e.g. without fail) is from late 13c., from O.Fr. faile "deficiency," from falir. The Anglo-French form of the verb, failer, came to be used as a noun, hence failure.
I just don't know which PIE form of "false" is the right one.
.jpg){kind=link}
Best Answer
I asked you which IE languages you know - in order to critically evaluate these three hypotheses, a strong background in the history of Latin (at least) is necessary. There are three major textbooks on the history of Latin - Baldi 1999, Sihler 1995, and Weiss 2009.
The first hypothesis is best supported by evidence - and, in fact, pretty standard now (for example, de Vaan 2008).
The Anlaut (word-initial) PIE *gwh> Lat. f sound correspondence is well documented, cf. Latin formus 'warm' - MnE warm; Greek thermos; Rus. zhar 'heat', goret' 'burn' etc. We still don't really know how PIE *gwh turned into Latin f (via *χw?) but this correspondence is regular.
We may ignore the perfectum fefelli because it's a relatively new coinage (double ll), cf. pello-pepuli, fero-tetuli (Meiser 1998), although reduplicated perfectum is usually archaic/rare in Latin.
The second "hypothesis" does not stand to scrutiny - supposedly, Latin fallo is derived from Latin facio. The person who came up with that hypothesis doesn't know Latin morphology at all. I don't know of any rule of Latin word-formation that could explain such a connection.
The third hypothesis does not have any explanation - it stops at Latin fallo.