Meaning of "reading law"
Some rightpondian can correct me if I'm wrong, but I think "reading" in this context is British for "studying", and it does usually imply university study.  The person in question is most likely attending the equivalent of law school.

In the US, lawyers typically write "Esq." after their name, which doesn't really signify anything specific but is traditional.  This doesn't have anything to do with degrees they may or may not have.  (And even lawyers with a J.D. degree don't usually write J.D. after their names, nor use the title Dr.)