Electronic – Replace fixed to variable potentiometer on DC regulator

No, grounding (to mains earth) is not necessary - the cell phone is not referenced to earth under normal circumstances, so using the (isolated) bench supply will be just like using a battery.

You need to set the voltage to whatever the cell phone expects (i.e. the nominal battery voltage) If it used a Li-Ion battery then the 3.6V setting should be okay.
The current setting just needs to be set to anything above the phones current requirement, the phone will only draw what it needs. If you want to be extra cautious, you can raise the current gradually to just enough for it to work (i.e. equal to the operating current, not above) Then if there is a fault (e.g. short) you will be able to see it (voltage will held lower than normal) and it is less likely to do damage.

For the exact operation procedure for your supply, read it's manual thoroughly (if you need more help with this bit tell us what exactly you are unsure of and provide a link to the manual)

Unfortunately, this PCB is unlikely to ever work. You will need to redesign it.

Switching power supply designs are very sensitive to layout. In particular, it's critical to minimize the area and inductance of the current loop. Unfortunately, your design places the inductor rather arbitrarily, which leaves the current loop all over the place. For more information on these aspects of design, you may want to read some application notes on switching PSU design, such as:

• Linear Technology: PCB Layout Considerations for Non-Isolated Switching Power Supplies

• TI: Five Steps to a Good PCB Layout of a Boost Converter

I also see a couple of generic issues with your PCB design, including:

• Your layout is not making use of its ground pour. A good ground is critical for all circuits, but is particularly important for a power supply design.

• One of the terminals of C1 appears to be unrouted.

• As other users have mentioned, it looks like you may have chosen some parts which are not rated for the voltages you're trying to produce.

• You are using rather arbitrary packages and footprints for a lot of parts. For instance, you're using a Kelvin resistor footprint for RSENSE1, but ignoring the kelvin terminals, and it almost looks like you've soldered an electrolytic capacitor to a SMD capacitor footprint on the back side. Switching power supplies are sensitive to lead inductance; these choices matter!