They didn't. Bruce trained with dozens and dozens of martial arts and other masters of various disciplines, including ninjas, and other experts from every corner of the globe. I can't see having the League of Assassins train him as an improvement over his comic book origins. Its kind of just a watered down version of it. It works for story purposes, but it pales in comparison to his actual origins.
Bruce later encountered Ra's Al Ghul, and at one point later in their encounters, while they were working together, Batman actually trained the League of Assassins.
All this is based on the comic book Batsuit, which has been described as layered Kevlar armor and mail since about the late eighties/early 90's. Jean Paul Valley used a type of chainmail armor when he took over for Bruce during KNIGHTFALL after Bane broke Bruce.
I don't know that I've seen razor sharp launching gauntlets in the comics, but the gauntlets and fins have often been shown to be a defensive/climbing measure, and various versions of Batman have had launching projectile bat weapons before BEGINS/TDK came along, including Jean Paul Valley in KNIGHTFALL, who fired bat-shurikens from his gauntlets. BATMAN BEYOND also featured these, and I think Batman kept them around for a while before TROIKA if I remember correctly.
Yes, but its one that, other than visually, is born of the comics, and even visually owes some inspiration to them. The Batmobile has always been a supercharged roadster, and that's what The Batmobile has been since the late eighties/early nineties, a combination of sportscar and fighter jet. There are even versions that have been more tanklike in the comics, as well as Batmobiles that jump, fly, etc.