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Yes, XnView MP can do all these with ease.
1,2,3) Just be sure to set up that the catalog will be on your shared drive and both instances from the two different computers will point to the same catalog.
4) Yes - I think that this is enabled by default. If not, see Tools|Settings|Metadata - all the checkboxes with „Export...”. RCK are saved in the file itself (if the format supports) - if not they are saved in an XMP sidecar. This is the industry standard and all the DAMs work in this way.
5) Sure, use Category Filter and Ctrl+Click on the desired Categories (this is how tags are named in XnView MP).