Would it be possible to send money through the Lightning Network to an offline address? My understanding is that the current Lightning Network spec requires an address receiving money to sign things with its private key - which requires the owner of that key to be online. Are there any techniques that would allow some 3rd party to accept money on your behalf without trusting that 3rd party?

I can imagine a situation where the sender delegates sending to a 3rd party, just like posting anti-cheat transactions can be delegated to a 3rd party that watches the blockchain. For example, maybe the sender gives a 3rd party the proper transactions needed to authorize sending lightning btc to the destination once the destination comes online. I don't know if this is possible with the current functionality in Bitcoin, tho. Is this possible?
This sounds like a web wallet, where the keys were managed for you and you simply interact through a website.  You'd have to trust the third party, but as we've seen on the main network, this is acceptable to a great number of people.

Just like with the existing Bitcoin network, you have to run things yourself to be trustless, but there are more convenient options if you are willing to trust (and pay fees to) other parties.