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About your second point, I just tested it on my machine, Word 2016 saves and opens *.odt files just fine. Is there some unsupported stuff?

Yes, their implementation is (deliberately?) patchy with many bugs. They've been lobbying to prevent governments from adopting ODF (an open ISO standard with multiple implementations): https://www.theregister.co.uk/2014/02/22/microsoft_uk_odf_re...