Headphone symbol when headphones not in use
The headphone symbol is shown when you are connected to Bluetooth headphones/speakers, which would be true if you have a Bluetooth device auto-connecting (which would not be used when you plug in headphones). Go to Settings → Bluetooth and see which device you are connected to, and disconnect if you wish. Sound notifications will be routed to this device so that is why you do not hear them.

If you no longer want to use Bluetooth to connect to these particular headphones/speakers (i.e. you only want to use a cable) you can also forget the device in the same settings by tapping on the info symbol  and selecting Forget this Device. This will stop your iPhone from auto connecting to them. If you decide in the future you'd like to use Bluetooth again, you can pair your headphones to your iPhone again later.