you don’t need any telemetry setup to run pi & navio. …Well, somehow you have to get a signal from your controller or groundstation to the Navio & pi.
You can pre-program everything and then let it do its thing but if there is a problem there is no way to take over control.
However, the rpi has got WiFi build in and given that you have access to a WiFi router and a computer you can establish a link in that way. - All described in the Navio setup instructions.
As to question 2: You can try to establish a link using PS4 controller with your groundstation (Mission Planner or Qgroundcontrol - both are free). Once a setup is created you might then be able to control the drone that way. But I’m not sure about Bluetooth support but do know joystick control is possible with mission planner.
In regards to your idea of being within 10m range: Bad idea, …during the first few flights you should ensure everything is working as expected and your drone is being tuned to your specific setup / configuration. 10m is not much space if a fast flying drone is suddenly “going crazy” and flying towards you. You can get nasty injuries that way.
(Even a drone like mine can easily do 50 km/h that is roughly 15m/sec - and it is not even a racing drone)
This also relates to your idea with the bluetooth connection as this is usually much shorter range than WiFi. (Usually less than 15m compared to WiFi at around 40m or more)
This brings us to your 3rd. question: You might be able to use the build in Bluetooth connection of Rpi, but you will have to find a software or a script (may have to write one using ROS package) for that. But once again you only would operate within approx. 15m using Bluetooth. …may be suitable for indoor flying but not really outdoors.
A much easier and effective setup would be using a USB Joystick connected to a Laptop or PC running Mission Planner and then using your WiFi to link to the Navio / Pi.
Laptop would be better as you can take it outdoors and see your drone easily whilst controlling it.
Another option is loading Qgroundcontrol on an android smartphone or tablet and using a screen joystick (touch control) to control drone over Wifi.
Whichever way you choose would highly recommend setting up fail-safe function like RTL if drone flies outside range of Wifi.
…This should also answer your question 4.