Query posts from different categories in multisite

For restricting the categories, maybe the plugin Restrict Categories can help. But it doesn’t have Multisite capabilities.
Take a look at the filter list_terms_exclusions that it uses in the cat_filter function.

But Role Scoper states that it has “Extensive WP-mu support”. May be worth a try.

For the query part, take a look at this Q&A, find a random blogid across my multisite network that has at least one post published.
Basically, you need the functions switch_to_blog and restore_current_blog.

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