I tried downgrading my graphics settings in Overwatch from Epic to Ultra, and it resolved all of the issues with the disappearing crosshairs and incorrect portraits.
So I suspect that one or more settings under Epic is what causes the issue, but I haven't experimented to pin-point which settings those are exactly.
Not sure about console but on PC at the top of your friends list there is a 'Recent players' tab.
If you right click on a person you will see an 'Add friend' option in the dialog menu which, provided that they accept the request, adds them to your battle.net friends list.
You can also right click on a person's name in chat and add them that way (as pointed out by GnomeSlice below).
Best Answer
This has been added in the November 16, 2017 patch.