What is the etymology of the Latin suffix "-ālis" (and related forms like "-āris") as in "nātūrālis"? Do we know any corresponding suffix in other Indo-European languages?
(Part of the info in this answer is from Buck's Comparative Greek and Latin Grammar, which, though old and outdated in many respects, is probably still reliable on this point.)
Ultimately, Latin -āli-(s) goes back to a PIE suffix *-lo-, with reflexes in many branches.
Latin -āli- is originally the suffix -li- added to a-stem nouns: e.g. animā- : animālis. It became analyzed as a single, productive suffix, however, and was extended to other types of stems: e.g. mort- : mortālis.
This -li- is the same suffix seen in adjectives like similis, humilis, etc. But it seems to have been originally *-lo-: compare the Greek cognates of these words, ὁμαλός, χθαμαλός. Transfer of adjectives into the i-stem class was pretty widespread in Latin (not just from o-stems but also from other stem types, cf. e.g. suāvis: Gk. ἡδύς).
As for PIE *-lo-, its reflexes form participles in Slavic and Armenian, and are also found less productively but still commonly in Greek (see the examples above and many others, e.g. φῦ-λον, τυφ-λός), Sanskrit (e.g. bahu-lá- 'abundant'), and probably other languages.