Hi!
Can it all be done in SPARQL against some services that already expose WP subcategories given a specific category ? Or is there an API that does this already ? other tools that might expose WP categories ?
I don't think subcategory relationship is not recorded in Wikidata. E.g. https://www.wikidata.org/wiki/Q7361750 contains https://www.wikidata.org/wiki/Q14436424 but neither have any indication of that. The problem I guess is that category hierarchy is different on all wikis, so it's hard to have one property that expresses it on Wikidata.
You could do through "subclass of" and "category's main topic" but not sure that'd capture all. E.g.: http://tinyurl.com/h7qpcdn but that only captures one subcategory, since other items don't have the same hierarchy in Wikidata.