What does “de-bearding” mussels mean

In order:

1. No. It's possible, though not especially common, for a dead mussel to be stuck closed; they'll likely open with very little resistance.
2. Most likely yes. The main issue with eating dead mussels is that it's an indication they've been poorly cared for; you can't know how long they've been dead or how fresh they are. In your case, you know with reasonable certainty that they were alive until very recently so they're still fairly fresh.
3. No. Mussels open when cooked in order to "breathe"; the excessive heat overwhelms them and they're trying to vent a little bit (which cruelly just seals their fate, but we as cooks benefit). If they're dead, they won't have this reflex and they won't open on their own. As alluded to above, most of them will be partially open already.

I would still cook them, but do a couple things to be as safe as possible:

1. Cook them without further delay, within a few hours if possible. Any dead mussels will start to go south faster than live, since they're no longer respirating.
2. Discard any mussels that don't open after cooking. Chances are these were among the small contingent who stayed closed after death and you don't know exactly how long they've been toast. Better to play it safe and accept the loss; besides which, it's a bit of a chore getting them opened up when you've got plenty of their conveniently open brethren ready to eat.

The mussels are most likely fine.

1. If they are visibly alive, it means that they spent the drive in an environment which was not harmful enough for them to die.

2. Even if some of them died during the drive it must have taken them some time to die. They spent the rest of the time in an air-conditioned car. Even if they died very soon after departure, they only spent an hour or so in conditions which were still okay for some of them to survive.

Do not listen to roetnig's advice about closed mussels. It is a well-known myth that they are necessarily bad. Wikipedia article on Mussels says:

Mussel shells usually open when cooked, revealing the cooked soft parts. Historically, it has been believed that after cooking all the mussels should have opened and those that have not are not safe to eat and should be discarded. However, according to marine biologist Nick Ruello, this advice may have arisen from an old, poorly researched cookbook's advice, which has now become an assumed truism for all shellfish. Ruello found 11.5% of all mussels failed to open during cooking, but when forced open, 100% were "both adequately cooked and safe to eat."

The article linked on Wikipedia is not a scientific paper though. I hope someone will do a more thorough analysis of closed mussels and their safety, but the popular rule of thumb seems to be pretty false.