As I'm sure you know, one of English's "small clause" constructions consists of a subject and a gerund phrase, where the subject may be either in the objective/accusative case ("them leaving was a surprise") or the possessive/genitive case ("their leaving was a surprise").
This is the case even when the subject is a dummy it. Hence, we find the following (real) examples:
- […] the reader is conscious of its being John the Baptist who speaks these words: […] [link]
- […] the possibility of its raining […] [link]
- […] most philosophers are committed to its being impossible that 1 + 1 = 3 […] [link]
though in all three cases, it could have been used instead (at the risk of irritating prescriptivists).
That said, I don't think this use of dummy its is ever possible with nouns as opposed to gerunds; for example, although "them/their leaving was a surprise" can be reworded as "their departure was a surprise", "its raining was a surprise" obviously can't be reworded as *"its rain was a surprise".
Consequently, I also don't think that the non-subject uses of dummy it have corresponding uses of dummy its. English does have an "objective genitive" — consider e.g. "her nomination", meaning "her being nominated" — but it's comparatively restricted, and I don't think it ever works with gerunds, only with nouns. (The nouns can be identical to gerunds, as in e.g. "his killing at the hands of […]", but I think they do have to be nouns: cf. "a series of killings".)