How do I “get the next 10 posts after post_id == x”?

Ha… While this makes a lot of sense, WP is not to good with arbitrary conditions like this and nothing like is implemented in WP_Query currently.

However since your issue is really chronological rather than requiring dealing with IDs, you probably can make use of Date Parameters to restrict results to be older than time of oldest post you fetched instead of explicit offset by count.