Search for 'isolation routing' - that should give you some sites to explore.
If you are designing a commercial product and want it to be ROHS compliant then you would need to use lead free solder.
In the small ground pour you can see a number of vias which I suspect connect to a much larger ground pour on another layer (bottom if that is a two layer board).
It needs to be the other way around. The pads on the mounting holes should be connected to the FG net so the ground pours on both sides of the board don't connect to the mounting pads. All component grounds should be connected to your ground pours which shound be the GND net. I would guess that only...
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Component manufacturers seldom put multiple ground pads that don't need connected.
If you look at the evaluation board document you will see that they use a ground pour to connect all the ground pads together.
So yes, all the ground pads should be connected together on your PCB.
I think you will find that FG stands for frame ground and will be connected to the pads that form the mounting holes. For EMC reasons you will often find that the electronics ground will be connected to the metal enclosure through the mounting holes and by joining the frame ground to the electronics...
Go back to your PCB design and select Edit->Select->Outside Area. Draw a box around your board then press the delete key to remove any track fragments outside your board outline.
No try re-generating your gerber files.