"He's good enough to take down the Joker with ease which Batgirl... both versions.. were unable to do."
Straight h2h or the kind of fight where the Joker plans and makes it work.
"But on the contrary, he's not good enough to take down lady shiva, which ONLY batgirl and batman have been able to do. He goes toe to toe with Deathstroke in a pretty even match every rip, Slade usually wins though"
Batman never won a straight figth against Shiva.
Against Slade, here is how Batgirl dealt with him.
http://img306.imageshack.us/img306/9941/060cy.jpg
http://img306.imageshack.us/img306/7825/071in.jpg
http://img169.imageshack.us/img169/4892/083ga.jpg
http://img83.imageshack.us/img83/9648/099qp.jpg
http://img170.imageshack.us/img170/9794/109ck1.jpg
http://img25.imageshack.us/img25/7725/113nb.jpg
About Connor, he has less h2h showings than Grayson, but the ones he has easily get him above.
Batman aint equal to Shiva btw. Thats Dragon and Tiger.