Does this mean it's possible to play with my device's clock and
manipulate the game time?
Yes. There is no punish for this as I have done this before to test the theory.
Your items will sell out, you will make gobs of money, and build times would be shortened (obviously).
Does the game try to "punish" you at all for doing this?
Yes, since version 2.0, when ever the game detects that the time was moved backwards, one of the bitizens will be lost due to "a horrific time traveling accident". The bitizen is chosen randomly, so It can be a bitizen occupying a dream job.
Also, be careful of when you chose to revert the time back. If you build / start to stock items and then revert back to the correct time, it will show the total stocking time = stocking time + time skipped ahead.
To avoid this, do not stock items and then move back to the correct time. Let all stocking finish, or skip the time forward till the stocking is completed.
One other issue I seem to have was that no more customers will come by your stores....
This is fixed by closing out the application completely and reopening it.
Best Answer
The game actually does backup - it backs up to Game Center.
From the app description - "Game Center integration to let you see your friends towers, back up your game and earn awards."
I have used the Game Center backup myself - I deleted the app for a while when I got bored of playing it, and then I reinstalled it after a couple of months. When you restart the game and sign in to Game Center, a message will pop up alerting you to a saved copy of the game (if there is one) that you can choose to use instead of starting over.
You can use iCloud/iTunes as well to backup your game. A post on the GetSatisfaction forum from Dani Rudolph, a Nimblebit employee, suggests following this Apple guide that helps you through the backing up and restoring process.
If you use iCloud/iTunes to back up, you can then use this Apple guide to transferring the information from one iPhone to another.