I have a friend who has an HTC One, and we had a hard time getting it to work. I think one reason is the HTC One's NFC sensor is near the top of the phone, while the S4 is near the bottom. I didn't notice any icon, except when transferring files. I had better luck connecting to a Motorola Razr M, so I know that it does work. Also, you may want to make sure bluetooth is enabled, since some files are transferred that way.
As for trying to pair your phone to your car, you probably need to delete a device from the car. I had to do this with an old phone no longer in use, and then I was able to pair my new phone.
I hope you're able to get the NFC working. I bought some tags which work, and I'm trying to think of cool ways to use them.