Steam client doesnt allow chat copy/paste operations. So paste web-links is possible only via browser version of CH. If seems strange because via console it is possible to fill buffer by commands like "partyanddecksbbcode". ADDED: subject actual about keyboard hotkeys ctrl+c/ctrl+v.
I thought it's some "feature" of Adobe Air or something. It would be great, if there's fix for it. Anyways, right-clicking and selecting copy / paste works just fine in Steam client as well. It's just the keyboard shortcuts (ctrl + c / v) that don't work in Steam.