What are the Consequences of Updating a Minecraft SMP Server?

Solution 1:

Will upgrading the server affect any of the currently explored map, including any builds we have put together?

No, all of your world that you have explored and all your buildings should remain.

With a different version comes unique map generation. Will there still be continuity in the map as we explore out further? Am I going to have a huge mountainside clipped into the ocean because of different map generation code or a snow biome immediately beginning on a newly explored chunk?

There will not be continuity. You will almost certainly have mountains clipped into oceans and the like.

Will any of our inventory be affected?

Possibly, but backing up the [Playername].dat files in your world folder should protect against that. Should anything happen to your inventories, copy your backups over the existing files.

Another thing you should be aware of: Every client will need to update also.