You can run a script with BigFix, output the results to a file, and then read those results with Relevance with an Analysis Property.
You can also get most information through Relevance directly.
Related: BigFix Documentation & Resources
You could technically download things and then move them somewhere and do nothing else, but that would require that you make a copy of the Fixlet or Task and modify it to do this. It would be possible to automate it through the REST API.
Sort of. You could change where the client itself is installed so that it is running from a volume with more space, but that doesn’t sound like the issue.
It seems like you want to install updates from a network volume because none of the volumes on the server are sufficient. This is technically possible but would require you to rework everything yourself. I don’t think you could do this very easily with the built in Patching Fixlets without modification.
If you are willing to take on complete automation of everything yourself, then it should be technically possible with BigFix.