Where does the $10 come into play? Can they only transfer up to $10's worth of currency?

From what I understand--yes. They would need to "settle" their transfer on the blockchain and then open a new channel. The solution to this from what I've read is an IOU mechanism and a third party that has a channel with almost everyone. If the third party is rich then each channel is theoretically huge.

ALICE --(IOU to bob for 10k)---> CAROL --(IOU from alice for 10k)--> BOB

I'm a bit of a BTC novice here, so help me understand something: Does this not mean that on the LN there will necessarily be BTC that's essentially "floating" in the network, that hasn't settled on the Blockchain? What does this mean for liquidity and the integrity of the network in general?

Yep basically. BTC is tied up in channels and when you want to take it out of the lightning network you write a transaction to the actual block chain. Though with the way lightning routing works you can keep your money in the network for months or even years making hundreds of transactions. It's kind of like having a debit account (bitcoin lightning) and a savings account (bitcoin stored in cold storage) with your bank.

What happens to the engine trustless selling point of bitcoin when the only plausible scalability solutions entail reintroducing trusted third parties?

The great thing about lightning is that it doesn't introduce trusted third parties: it only requires that nodes have commited funds.

You don't need to trust the third parties, that's the point.