Plug corded light fixtures in outlets in ceiling per code

Originally, in the UK after BS1363 was introduced, there were 1, 2, 3, 5, 7, 10, and 13A fuses but nowadays mostly 3,5 and 13 are used/available.

The plug fuse is to protect the wire between plug and appliance, it is not primarily intended to protect the appliance (which can have it's own fuse or other protections).

So the choice of fuse principally depends on the diameter of the wires between plug and appliance.

However there is nothing wrong with choosing a lower fuse capacity that will also protect the appliance to some degree.

In most cases, simple switches are equivalent. They break the circuit and do nothing else. So switching (breaking the circuit) further down the line is usually fine.

There are some exceptions, such as dimmers, multi-position switches, and multi-pole switches, none of which seem in play here.

The computer is different because it need to be shut down (program exited) rather than simply turned off. And some computers never really turn off but keep a very low power program running to allow fast start or remote update or activation.

The use of LED bulbs in a socket that is properly rated for incandescent bulbs will likely not generate any dangerous heat when properly mounted. ETL seems to be similar to UL safety listing. You do need to check if the mounting instructions specify minimum clearances.

The only risk you may face is possible damage to the finish of the vanity after long term grip by the rubber coated clamps. Some rubbers and plastics break down or give off solvents over time.