No he can't do that from a ping. But if he's on the same network as you (like for example in a different room and you are both sharing the same broadband connection) then he could in theory run a "packet sniffer" and see all the traffic going to and from your computer (content of emails, web sites you visit etc).... if he knew how. This is only really possible on a local network though.
Dh is a software engineer so I'm sure he could do all these things you mention, but we don't have a LAN so at least I can't be put on-the-spot! Anyway, he knows about my addiction . I'll just have to make sure that I do visit a few other sites so that I can claim 'Oh, yes, I did visit Mumsnet briefly, but most of the evening I was researching the new xxxxx' [insert 'innocent look' icon].