Understandig of boost converter
"why is the voltage increase with the duty cycle?"
Simple: if you increase the DutyCycle, you charge the inductor for longer, hence it contains more energy at the end of the charge. On the discharge cycle this energy is transferred via diode D into the load and the capacitor.

Yes ILmax will be higher because it will be charged more. Realize that a capacitor is charged with current and the voltage shows how much it is charged. With an inductor it's the other way round, it is charged by applying a voltage (this happens when S closes) and the current shows how much it is charged.

You focus a bit much on Vl, the voltage across the coil. But that is not so important, the inductor current is what matters. The inductor behaves as a current source when it still contains charge and S is open.

Note that for the same dutycycle if you increase the value of the load R, the output voltage V0 will increase ! A given Dutycycle does not result in a constant voltage at the output. So that's why real boost converters need a feedback circuit to control the dutycycle.

If you would remove load R, the voltage would increase to infinity ! (in theory that is)