I agree that the individual mover is as, if not more, important than the company. Finding someone local to the beginning of your move, that you can check out with references, (maybe check Craigslist?) and that you can meet beforehand will likely be the best plan.
The driver will likely be the loader/mover for your stuff and will also unload on the other end. Having a chance to meet him ahead (not often easy), make a connection on moving day, having his cell number for contact *during* the trip, and discussing thoroughly your concerns before he supervises the loading, is the way to head off many problems.
Provide drinks, a sandwich lunch, a sweet treat, and plenty of paper towels at the kitchen sink for wiping sweaty brows and hands.
Usually refrigerators are not moved. I use them to store all the "extra" items I'll need during loading day that I don't want anyone to pack up by mistake. :D
Jan in VA