Can you send amounts that would typically be considered 'dust' through the Lightning Network?
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.