What is the ultimate etymology of "false"?

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.

Here's the OED's etymology:

Etymology: late Old English fals adj. and n., < Latin fals-us false (neuter fals-um , used subst. in sense fraud, falsehood), originally past participle of fallĕre to deceive; compare Old Norse fals n. The adj. is found in Old English only in one doubtful instance (see sense A. 13); its frequent use begins in the 12th cent., and was probably due to a fresh adoption through the Old French fals, faus (modern French faux = Provencal fals, Spanish, Portuguese, Italian falso). The continental Germanic languages adopted the word in an altered form: Middle High German valsch, modern German falsch (compare Old High German gifalscôn to falsify), Old Frisian falsch, Dutch valsch, late Icelandic (15th cent.) falskr, Danish, Swedish falsk.

The etymological sense of Latin falsus is ‘deceived, mistaken’ (of persons), ‘erroneous’ (of opinions, etc.). The transition to the active sense ‘deceitful’ is shown in phrases like falsa fides ‘breach of trust, faithlessness’, where the n. has a subjective and an objective sense. In mod. English the sense ‘mendacious’ is so prominent that the word must often be avoided as discourteous in contexts where the etymological equivalent in other Germanic languages or in Romanic would be quite unobjectionable. Some of the uses are adopted < French, and represent senses that never became English.