You should be able to call eg https://developer.atlassian.com/static/rest/stash/3.10.2/stash-rest.html#idp1759552
They are paged APIs, so you can use a small page size, eg 1, and it will give you the total number of projects found in the "size" key in the response. Same for the others.
EDIT: Actually I just tried that, and it gave me the size of the page I requested, which is a bit pony. I guess you need to page through the entire lot to get the total.
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