Should I trust movers with my equipment?

Movers yes, shakers no.

There is nothing certain in someones else's hands that I wouldn't entrust firstly to my own.

Better to inquire first about the reputation of the moving company. I don't think a company has years of experience could do this type of silly mistakes. What I think first you should prepare a list of whole things and keep it to you it would help you after the move to check out the status of each stuff. I would suggest better to ask prior, I mean who would pay the loss in case of any damage.

Rule no. 1 - Expensive electronics have about as much chance of making it all the way across country in one piece as a camel has of passing through the eye of a needle.