What is the study of language usage types [a question, an answer, a criticism, a complaint an elaboration, etc]

Neither 'ontology' nor 'typology' are correct. 'Ontology' is a sub-discipline of philosophy and 'typology' is a linguistic discipline concerned with types of languages (not language use).

Nor is 'conversation analysis' the right answer. Conversation analysis - as the name suggests - is concerned with the analysis of conversation. As such, it is concerned with only a subset of spoken language usage. It has made an enormous contribution but does not deal with language as a whole (it also was not initiated by Austin who only has a tangential connection to it - but rather Garfinkel, Sacks and Goffman - the first outlines appeared about the same time as Austin's work).

The closest you get to a typology of language usage that covers all the things you list would be stylistics (building on the traditional discipline of rhetoric). However, the shortcoming of stylistics is its focus on written language and non-spontaneous speech. So between it and conversation analysis, you will find a good coverage of the issues. Stylistics is also tainted by its frequent normative / prescriptive focus.

When it comes to a typology of broader types of language usage, corpus linguistics has made an important contribution through its efforts to sample and represent language across all modalities.

However, it must be admitted that there is no subdiscipline of linguistics whose aim is pure typology of language usage. (Even the new 'usage-based' linguistics is not concerned with that.) Therefore most approaches to language use some sort of ad-hoc typology (often drawing on concepts from rhetoric or stylistics without deeper consideration of their foundations).