Electrical induction at a distance through coils for an LED night light
For anyone interested, this YouTube video is a good resource for doing what I was asking. It shows you what you can achieve in practice, and is enough to answer almost all of my questions.

https://www.youtube.com/watch?v=4UmVLfFNx7U

It is completely feasible, you can do it safely with a simple transformer at typically low voltages and amps (9V to 12V @ 1A to 2A), a few coils, and a few LEDs. I'll post results after finishing the project.

EDIT:Another similar video:
https://www.youtube.com/watch?v=Kj0dq8m_iBc