So how does the system know what time it is? Host time is not reliable. Genuinly curious.

This is the reason why Ethereum contract usually set start time, end time and similar in block numbers. Block number is reliable while timestamp is just a local time of the miner that mined the transaction in which the contract call was included.

Timestamps in the block header. These timestamps are used to regulate the proof of work to maintain a block interval, so in the end both measures have the same time accuracy.

This isn't quite accurate to how we set it up.

Each step in our protocol could set a different amount of delay to close the channel. Beginning the close process does not require waiting, but finishing it does. It's actually the opposite, if you wait too long to close, you may not be able to close at all. This means that if you need to close it, and you can't reach your peers for the honest_close case, you better get to it!

Because each step can pick delay, the honest_close case allows a channel to close without any delay.