Can you send amounts that would typically be considered 'dust' through the Lightning Network? Reading through the Lightning Network Paper, it seems that amounts sent through an intermediate node use the Hash Timelocked Output construction. The thing I'm concerned about with HTLCs is that they actually require their own output. Simultaneously, the transactions they are in must also be broadcast-able if the other party goes offline or become uncooperative.

So, let's look at a scenario where Alice wants to send 1 satoshi to Carol, through Bob, with whom Alice and Carol both already have payment channels open. Alice sends bob a new commitment transaction with 1 satoshi deducted from her balance and a new HTLC output for 1 satoshi. Bob sends the same type of channel update transactions to Carol. The problem is, though, that HLTC output would be considered dust if it ever did actually need to be broadcasted! So this cannot be considered a valid guarantee of payment, since the transaction would not even propagate on the network.

Is that a problem? Part of the excitement over the LN is that you can send really small amounts, but it seems to me that you can't actually send smaller amounts than you could in a real bitcoin transaction, because you have to be able to broadcast the transactions if the other party doesn't cooperate or goes offline.
You are correct that small payments would be a liability in the case of uncooperative nodes in a channel.  I also believe you are correct that when the values get small enough, the trust-less nature of a channel breaks down.  Although a 1 satoshi transaction is possible, at some point the cost of the transaction that closes the channel is higher than the value transmitted over the channel.  You couldn't open and close a channel just to send a single satoshi, even if both nodes were cooperative.

What the LN really does is create a network similar to the existing network of correspondent banks in the legacy financial infrastructure.  With every correspondent banking relationship (payment channel, in the LN), there are risks and liabilities involved.  These risks are mitigated by charging fees to use the channels, which is exactly what the Lightning Network will do.

I don't see this as a show stopper for the implementation of the Lightning Network, but it will be a factor in what drives the fees for using a channel.  This dynamic may also play a role in how the nodes split the money that is originally locked to create the channel.  Less established nodes may have to contribute a higher percentage than those nodes that have been deemed trustworthy.  I'm speculating here, but this is one potential way to mitigate risks.

Just like with Bitcoin, the system should work so long as there are more honest nodes than dishonest nodes.  In this case, however, dishonest nodes must attain enough money to disrupt the network as opposed to hashing power.