John/River
Yes, well, it's in his nature to be a nosy bugger, isn't it? There is nothing untoward to what you said, only an expression of discomfort in the degree to which fans for your fictional self are invested in your relationship with Sherlock. It's an entirely understandable reaction to have.
I don't see why not.
[OOC: Rather than taking the time to mess around with it, River just uses her sonic screwdriver to hack into the Network and make the conversation private.]