I read in this article that Wally West is the 3rd Flash. I also read that his father is named Rudolph. This got me confused. As someone who has never read The Flash, is this some different Wally West? I mean, I thought Wally was Joe's son and therefore Iris' brother, certainly not nephew
No, the Wally West on TV is not the same as either of the Wally Wests that appears in the comics; then again, neither is Iris West (who's father is named William and didn't know Barry until they were both adults).
None of the characters on the show are exactly like their comic counterparts -- which, by the way, there are more than one. For most major DC character, there are four "main" versions (Golden Age, Silver Age, Post-Crisis, and New 52 versions) and dozens of "alternate universe" ones. The show, like all comics-to-TV adaptations, is taking pieces of the various comics canons and using them to build their own version of the characters.
We have no idea if Wally is ever going to end up a speedster. I seriously doubt they will ever call him Kid Flash, or that he will end up replacing Barry as The Flash (for one thing, they're the same age, while Wally in the comics is much younger.)