Does time pass on a Minecraft SMP server if nobody is logged on?

I have a private Minecraft server where we are 2 people playing.

Earlier today I logged on, observed it was night, and logged back off.

Then I logged on just now, it was day again.

In the meantime, I know the other player has not been online.

So my questions (they are related) are these:

1. Does time pass in Minecraft if the server is running, but nobody is online?
2. Can I use this to my advantage to:
   1. Cook or smelt things in furnaces while I'm offline?
   2. Plant trees or other things before I log off and log back on to them being grown?
   3. Get grass to spread while I'm offline?

This might be a related question, but if not, please just leave a comment and I'll take it out to avoid having a too broad question:

1. Does time only pass in a region if players are in the region? Like, if both of us are online, then I log off, and the other player is far away, does "time pass", like trees grow, furnaces run, etc. in the region he's not in? I would imagine that would take a load on the server if there are large explored areas.

Time in the sense of day/night cycle and weather passes while no one is on, however in all other senses time doesn't pass in a given chunk unless a player is online and nearby enough (although for a small set of chunks near the default spawn, "nearby enough" is anywhere in the Overworld).

Chunks do not update with no one around so I created a character called "updater" to just sit inside of my server. This will allow chunks to be updated, even if no one is on the server.

time passes, however, if the chunk is not loaded then you will not be able to use this to your advantage.

there are mods that can load chunks whilst you are offline