My familiarity with the Batman franchise is pretty much limited to live action stuff. I saw a few episodes of the old TV show when I was a kid (and maybe I just missed the subtleties, but I got the impression Catwoman was a villain like the Joker rather than a love interest as the films portrayed her), and all the movies since the Burton/Keaton films, and I read a novel about a post-apocalyptic Gotham city. Now I'm watching Gotham, and starting to get curious about who's an original character and who exists in other Batman material so as to understand exactly which characters are in any sort of danger. I mostly assumed that anyone with weird powers or gimmicks would be a supervillain in training, but some of them get killed. I also thought the mob bosses from the movies were just made up to represent ordinary criminals to serve as a contrast with the guys like the Joker & Riddler. Except Tom Wilkinson's & Eric Roberts' characters from "Batman Begins" and "The Dark Knight," respectively, have both shown up on the show. The Penguin has the same name he had in Batman returns, and there is a female thief called Selina, like in "Dark Knight Rises," who shares a name if not a profession or backstory with Catwoman of "Returns". There is a hitman in "Gotham' called Zaz, whose name was mentioned by Crane in "Begins", who apparently cuts himself when a target gets away. There was also a girl in the pilot episode whom a lot of online commentators called Poison Ivy, but who did not have the same name as Uma Thurman in "Batman and Robin".
Victor Zsasz is a serial killer who notches his own skin per victim.
The first Robin is Dick Grayson. Eventually, they have a falling out after which Grayson goes solo under the name Nightwing. Jason Todd is the 2nd Robin, who dies at the hands of the Joker. (quite brutally too I might add) The 3rd is Tim Drake, who showed great promise as Batman's successor. The 4th and current is Damian Wayne, the biological son of Bruce and Talia al Ghul (Ra's daughter).
There were 2 other Robins, both female. Carrie Kelley, the girl with glasses, is non-canon. Stephanie Brown was Tim Drake's gf and a costumed vigilante in her own right. She filled in for Robin when he resigned for awhile.
Ivy Pepper is not (and never was) a name associated with Poison Ivy. Her real name is indeed Pamela Isley.
Yes, they are canon. In fact, Renee becomes The Question later on. (masked vigilante obsessed with conspiracies) Cris becomes The Specter, a supernatural entity.
Falcone and Maroni, yes. Grissom was a Burton creation.
I honestly don't know. The Joker's origins have been written by different writers, all with varying backgrounds so no one really knows who the Joker really is or if any of his origin stories are true. I think Nolan hit upon this fact by having his Joker come up with different childhood stories.
Big big deal. Bruce's best friend and one of many sources of his guilt. Falcone hired someone to throw acid on Dent's face during a trial. Dent went insane, the rest is history.
New; created for the show.
No to Batgirl. Spoilers in white: Batgirl is Gordon's daughter.
Yes to your latter questions. He's supposed to be a former actor. He does have some training in the field. The 90's animated cartoon had him be a former MI6 operative or somesuch.
Rachel is a Nolan creation. Vicki yes. Poison Ivy's a super feminist, so no. Julie Madison (Elle), yes.
OTP depends on the alter ego. Talia al Ghul is in love with the man behind the mask. Batman and Catwoman are friends with benefits, although Selina does know his true identity.
I don't know any jokes. Or are you talking about the homoerotic subtext between the bat and the bird? No, there's nothing between them beyond the usual guardian/warden stuff.
Yes to Barbs, she is the original. Sarah was a Frank Miller creation that was retcon'd in the current iteration of DC universe.
Hope that cleared some stuff up.
I'm sure there are others more knowledgeable than me that can correct me if I'm wrong about anything.