I've been on both sides of this moving other people on personal, local, and corporate moves, had professionals move my stuff and moved myself.
The reason certain things are excluded is they require hazmat endorsements and papers to transport that most moving companies don't have. If they get caught transporting something as simple as a full propane tank they could lose their license and by extension their livelihood. I wouldn't carry your stinking kitchen cleaners either at the risk of my job. I'd rather give you $20 to buy new ones for your new house.
A well established moving service will have an established inventory and claims process to handle problems with your move. The movers will inventory everything they take and you should sign at origin and destination certifying the inventory is complete and everything is there. You could make a list of your guns with serial numbers included and compare it to the inventory they create at origin when you sign there, then unpack your guns at destination and check them before you sign off on delivery. If anything is missing or broken you can file a claim with the claims department and it'll be handled by a company with a national reputation.
It all depends on the company. The one that moved us moves a lot of Corp Exec, sports clebs & musician from the Nashville area. They were hired by my wifes company and we will be requesting the same crew the next time we move. Like all things its all about the company you choose.
I have to respectfully disagree with this. In many cases the same people work for many different companies on a day to day basis. Whoever has work from one day to the next is the company they work for. They work for owner/operators for cash under the table and sometimes do pretty well for themselves, at least over the summer. The better workers are hired full time by subcontractors that only work for one company. I don't know of any way to control who comes to you at scheduling time. You might see if you could request a contractor who works out of a warehouse instead of a regional or national driver.
On a happier note, we once put a Pontiac Montana minivan into a 40' shipping container to go to Europe. Up the ramps into a lower straight truck, then across the ramps from the straight truck to the can. Nerve wracking just to watch. I never saw anyone double stack cars, but I've heard it's not that hard, just time consuming and not the best use of space. Movers don't make much money hauling cars unless the trailer isn't full anyway. Not worth the weight for the space they take up if there's a chance of filling that space with another load or something more dense.