What's the relationship between "separate" and "eight" (old and modern meanings of 八)?
Generally speaking, we cannot create pictures or ideograms for every morpheme in a language; we'll quickly run into practicality issues if we try to do so. 「八」 (Baxter-Sagart OC: /*pˤret/) should be treated as just a rebus or phonetic loan for eight.
As an ideogram representing to separate, 「八」 either [represented a morpheme cognate with] or [was the original representation of] 「分」 (/*pə[n]/, to separate), with the component 「八」 serving simultaneously as a meaning and sound hint in 「分」.
If 「八」 was the original form of 「分」, then in terms of character development, 「八」 complexified into 「分」 via the addition of 「刀」 (knife, indicating the meaning to separate an object into two with a knife) after 「八」 was overused for the meaning eight, causing a need to disambiguate the two morphemes eight and separate.
In any case, it is quite likely that the original morpheme that 「八」 represented is now [also] written as 「別」 (/*pret/, to divide). The financial numeral for eight, 「捌」, still uses 「別」 as a phonetic component, showing the conservative nature of the sound of 「八」 for an ancient morpheme meaning to divide, differentiate, separate.

The good luck and fortune meaning (發財) is an unrelated wordplay, although it touches on the same kind of rebus principle. 「發」 (Middle Chinese: /pʉɐt̚/) sounds similar enough to 「八」 (/pˠɛt̚/) for this purpose, and in an alternate universe or a now-lost topolect, 「八」 might have been used as a phonetic loan for 「發」.