According to Miller (2006: 76, 78), the endings -men and -mentum form a deverbal (with one exception) noun with the semantics of means, instrument or result of action of the verb. Relevant quotations are
§3.4 -men [...] ‘means, instrument, result’
While -men is formally and functionally related to -mentum (LG i.§326), the
latter will be treated separately for reasons stated in §3.5. [...] Of the two, -men has ceased to be productive since its main function was taken over by -mentum. Most -men derivatives have full grade of the root, but a few have zero grade as well; see -mentum below and Schumacher (2000: 114 f.).
§3.5 -men-tum [...] ‘means, instrument, result’
Historically, -mentum is sometimes considered an enlargement of -men of
Indo-European date (LG i.§371). As mentioned in §3.4, numerous doublets
existed, such as reg-i-men/reg-i-mentum ‘rule’. Syntactically, however, -mentum shares more with Greek -μα-/-ματ, from <PIE> -mn̥-(-mn̥-t-)
(IEL 209), than with L<atin> -men (Sandoz 1994: 328). Apart from rare deadjectivals, such as rudimentum [...] (<from> rudis [...]’), most of the examples are deverbal.
To summarize and provide exact answers to the questions, the ending (I would rather not call it the suffix) -mentum
- mostly used to transform verbs, and a few specific adjectives
- into a noun capturing the semantic category of either instrument or result of the verb's action.
Speaking of your “pet theory,” I would not call it incorrect right off the bat. While it does not align morphologically with the state of PIE affairs in Latin, it has its merit: your examples quod arguitur, quod frangitur, quod docetur are essentially periphrases for the (missing) middle participle. Then, semantically the Latin paradigm quod X is close enough in its semantics to that of instrument or result of action. This is not so much a straightforward theory, but rather a way of roughly delineating the semantics of the ending -mentum. The quod construct is fuzzier, however: Is quod pingitur something being dyed (more of the passive sense), or something being applied as a dye (more of the middle sense)? Pigmentum is the instrument, an aligns more with the middle sense of the quod construction, while quod pingitur out of context would be less specific.
It is also notable that the doublets that Miller is speaking about (fragmen-fragmentum, documen-documentum, perhaps no less than 100 of these) are not observed of "older" nouns, continuously traced back to the proto-language (such as nomen, germen, semen, stamen, vimen, omen)—there is no e. g. *nomentum. This is a clear illustration of the fact that the ending -mentum was productive in Latin, while -men was not, and many of the -men words were probably back formations from their -mentum counterparts.
Miller, Gary D (2006) Latin Suffixal Derivatives in English and their Indo-European Ancestry. Oxford
LG = Manu Leumann et. al. (1977) Lateinische Grammatik, Munich: Beck.
IEL = Meier-Brugger (2003) Indo-European Linguistics, Walter de Gruyter.