Mario Kart Arcade GP actually does save. Some cabinets have a slot for what's called a "Mario Card" which works on both GP and GP 2 and allows you to save your progress as well as access other features, such as getting items as prizes in races for use in the final round of a cup. However, nintendont doesn't emulate the Mario Card yet, and probably won't until full Triforce compatibility is achieved. If you're on a Wii with Gamecube compatibility and you use Nintendont for most of your Gamecube games, you can always download Quadforce which is a cMIOS designed to emulate the Triforce games (4.0 for SD cards, 4.1 for USB) and it does emulate the Mario Cards. However, you'd need to access the games via Dios Mios Booter, and Quadforce replaces any MIOS or cMIOS already there.
As for the analog thing, as someone said before, sometimes the game bugs up in Nintendont. A good rule of thumb to determine if the analog is going to work is to look at the credits text and see if it jumps back to 1. If it does, everything's working fine. If it stays at 0, it's probably best to reset.