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.
Trickle-down
The standalone combination trickle-down is in the OED:
trickle-down adj. of or based on the theory that economic benefits to particular groups will inevitably be passed on to those less well off; also transferred as n., a filtering down (of money or ideas). orig. and chiefly U.S.
[1931 W. Rogers in Tulsa Daily World 12 July iv. 7/3 What about the old Boys here on the home grounds? Well maybe this thing will eventually reach him in some beneficial way. Lord knows what way it may trickle down to him some day.]
1944 Antioch Rev. Summer 192 In agriculture, as in business, they are devotees of the trickle-down philosophy.
1949 H. S. Truman in Sun (Baltimore) 6 Jan. 6/1 We have rejected the discredited theory that the fortunes of the nation should be in the hands of a privileged few. We have abandoned the ‘trickle-down’ concept of national prosperity.
Later quotations talk of a trickle-down tax program, trickle-down housing program, ‘trickle-down’ process, trickle-down economics and a trickle-down effect.
Trickle-down government
The earliest use I found of the specific phrase trickle-down government appears to be in the Official proceedings of the 1984 Democratic National Convention, possibly spoken by Dorothy Vredenburgh Bush:
(Applause) It is bad enough to have trickle down economics, but the real evil of this last four years is Republican trickle down government, and that is what we are going to change.
Another snippet that appears to be from a 1984 Empire state report:
... there some philosophical difference you have with the majority?
POOLER: We view the world differently. I think they engage in what I call "trickle-down regulation" to be analogized correctly to trickle-down government. They think if you help the companies, in so doing you'll be helping the ratepayers.
The 1990 George Bush: 1989 seems to contain the phrase 14 times in total, including:
He wants $150 billion in new taxes. He wants $220 billion in new spending. That is not change, that is trickle-down Government. We do not need any more of that. His numbers don't add up. Anyway, he says he's going to just sock it to the rich.
...
Now let's look a close look at what he offers. Chris mentioned it, $150 billion, this is for openers, $150 billion in new taxes; $220 billion in new spending. That is called trickle-down Government. It goes right from the top into your pocket. And we don't need that anymore.
...
And yet, Governor Clinton has already proposed $150 billion in new taxes and $220 billion in new spending. You talk about trickle down, that is trickle-down Government. Give the Government you wallet, man, and step back and let Washington solve the problem.
The phrase was used much more in 1992 by President Bush when accusing Clinton's policies, notably during the third presidential debate.
Best Answer
Government comes from the term govern. From Old French governer, derived from Latin gubernare "to direct, rule, guide, govern", which is derived from the Greek kybernan (to pilot a ship).
Don't believe the nonsense you read online. There is precedent that the suffix -ment is derived from the latin mente meaning mind in some languages, particularly Old French. Words deriving from the mente sense generally have the suffix -wise or -ly, and are adverbial in nature.
But, it is also from mentum - (instrument or medium). It is this second sense that was imported into English.
In English, -ment means: the means or result of an action. Per multiple sources -ment is derived from the Latin mentum via Old French. For example, the Online Etymological Dictionary is quite clear on this subject.