What are the reasons that babies like vibrations? Do vibrations help
babies fall asleep and why?  How is it determined what level of
vibration is safe in procucts, such as for a vibrating chair?

I think you will find that babies like movement, not specifically
vibrations. but including vibrations.
This is because for all their life before birth they are floating in a
"moving swimming pool". Everything th emother does or says creates
vibration. The sound of her voice, the constant beating of her heart,
the multitude of noises produced by the gastrointestinal system as
food passes along, and every breath the mother takes.
It's like a really noisy, wobbly place to be. 
Reproducing that comfortable familiar environment is soothing to a
baby -this is why a walk in the pram, or a ride in the car will often
calm a screeming baby.
A still quiet world is foreign and in need of exploring, a noisyish
moving one is more relaxing.