1. Those sorts of lengths shouldn't be a problem. However, I'm slightly confused by your use of the terms 'power supply driver' and 'power supply box'?
Really I'd recommend putting the driver as close to the box you showed a picture of in your first post as possible. The driver will have mains voltage going in to it from that box, so will need to be terminated in some sort of enclosure safely - a double plastic back box might be adequate if that location is inside and the driver isn't too big... You might even be able to alter the existing single box to a double one.
Then the cable you run to the lights can be the 12V cable extensions that come with it, or any suitable flex - as it's low voltage there is much less risk of shock so not as necessary to be sure that it's protected though you may want to keep it out of the way of foxes/squirrels etc who do like to have a nibble.
2 The driver will likely not have an earth connection on the mains feed in as it will be double insulated (logo of a square within a square) and does not need one. Any cable feeding it (if not within the same enclosure) should have an earth though which is terminated correctly at the supply end.
When you cut off the plug adapters, you will almost certainly find it's 2 core flex - the one you were looking at was probably mains voltage to the light fitting, with a transformer internally if LED, or direct to a 240V lamp if it was a GU10 type.
In a garden, it's generally much safer to have only 12V going through cables - and much less worrisome if there is a problem.
One final thought- you may be best buying one of the kits first to double check the driver requirements (it should be written on the transformer plug)
It's possible that it could be 12V AC, rather than 12V DC, which would require a different driver to the one you've mentioned. It doesn't seem to specify anywhere on the data sheet or descriptions I can find of the light.
Re the driver you did mention - There is always a bit of a risk with a random name driver from Amazon that it will either not be very reliable, or will not last too long - so I'd generally recommend getting one from a reputable supplier - some of the reviews on that model look a little less than stellar....
Sorry i should of said:
Power Supply Driver = Constant Voltage LED Driver
Power Supply Box = I have a light switch in my kitchen that is wired to a box that is in a porch that is inside my house, a room that seperates the kitchen from the garden, the box right now is being used to power up a security light in the garden, i will be using that same box to power up the LED Driver.
Hope that makes sense, i'm starting to confuse myself now ?
I have done another diagram that should make things more clearer.
I will try and find a box to cover the LED driver (still looking for one)
I have sourced another driver with the details below and a picture of it in the diagram, hopefully this one is better than the one from Amazon.
Mean Well Constant Voltage LED Driver 60W 12V, MPN: LPV-60-12
Brand: Mean Well
Dimmable: No
IP Rating Specifications: IP67
Operation Mode Specifications: Constant Voltage
Output Voltage: 12V DC
Product Type: LED Driver
Sku: LPV-60-12
Warranty: 3 Years
Wattage: 60 W
Key Features:
Input voltage: Mains voltage 90-264V AC.
Output voltage: Choose between 12V, 24V, 36V or 48V DC.
Constant voltage, suitable for use with LED tape.
Wattage: 60W max.
Current ratings:
Suitable for use with LED strips and other constant voltage LED lighting installations.
This power supply is ideal for use under kitchen cabinets as it has mounting holes and can be screwed directly to the cabinets.
Non dimmable, we also have a wide range of mains dimmable LED drivers
Water and dust resistant to IP67.
Guarantee: 3 years
Manufacturer: Mean Well
I will contect the seller to see if that transformer plug is 12v AC or 12v DC
Hope the above makes sense.