Unless it's a 1bedroom apartment don't accept less than 3 guys. Anything heavy really benefits from multiple people. Get three quotes for the job, talking to someone on the phone for each quote. Ask questions during the phone call. Ask about the company and the service they provide, treat it like a short interview. Select the company that sounds the most professional, unless their price is exorbitant compared to the other two.
When the crew arrives, let the leader of the crew know you'll tip cash for good service.
Letting them know you'll tip cash is the best advice here. The movers will treat your stuff better and stuff won't go missing (although I have no reason to think this is specifically a concern, I do understand the paranoia).
I moved a 3-bdrm house with 3 guys. Not sure why you'd say it's necessary to have more. It takes more time, of course, but if you're paying by the man-hour, that doesn't change the price. You definitely need 2-3 to wrap/move couches, but there weren't any items that would have benefited from 4 or more. It's just more parallel processing, which is nice but not critical IMO.
100% let them know about a cash tip, be generous too. They're handling all your stuff, if you want them to treat it right they've got to be incentivized to do so. It's a hard job. If it was easy money everyone would be doing it.
When you do tip, give each guy their share directly.
Also buy a bunch of gatorade and water or other drinks and put it in a cooler with ice and tell them to help themselves to it, especially if it's at all warm where you are.
Finally, don't be a hard ass and don't let them walk over you either. Don't get on their case about stuff that actually doesn't matter, and be firm where it does matter. You're paying them to do a job, but they're more likely to respect you and your stuff more if you don't treat them like crap (like some people undoubtedly do - they've told me).