I read in this answer that there may be a relation between the future endings -bo, -bis, etc. on the one hand and the verb fio "become" on the other. Is this true? If so, do we have any more details on how that happened
Bonus side question: are fio and facio related, cf. utriusque participle factus?
The verb fīō comes from (the zero-grade of) the PIE root *bhuH-, "to become". This led to a Proto-Italic verb *fui-. In Proto-Italic, the phoneme *f is thought to have been a bilabial fricative, voiceless [ɸ] at the start of a word and voiced [β] everywhere else.
Somewhere around Proto-Italic, this verb *fui- started to be used for certain periphrastic formations: *fuā- (subjunctive) for the imperfect, and *fui- (indicative) for the future. Then, just as in the verb fīō itself, the *u disappeared before these other vowels.
This left Proto-Italic (or Post-Proto-Italic) forms like *amā-fā-s "you were loving" and *amā-fi-s "you will love". In other Italic languages, this medial *f remained: Oscan fufans "they were" (= Latin *fubant = erant), Faliscan carefo (= Latin carēbō). But in Latin, medial *f [β] hardened into a stop b, giving the endings -bās and -bis and separating them thoroughly from the verb fīō.
See this paper for more information. This paper also points out that these tenses are also seen in Celtic, and that their origin is much less cut-and-dry than I make it sound.
Faciō "to make" comes from the Proto-Italic root *θaki-, probably from PIE *dheh₁ "put" (compare Greek tí-thē-mi, English "do"). I haven't seen a good explanation of the *k, but the *θ seems to be attested in other Italic languages. In Latin, *θ at the start of words later turned into f.
Fīō "to become" comes from the Proto-Italic root fui-, from PIE *bhuH- "become". This verb was rather defective even in Proto-Italic, and only had three tenses: the present indicative, present subjunctive, and perfect indicative.
Since fīō was so useful, and not a full verb, it was eventually "scrapped for parts" in Latin: its perfect forms (with the stem fu-) were stolen by esse "to be" to act as its past tense, while its present indicative and subjunctive forms (with the stem fi-) were stolen by faciō to act as its passive.
