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.
If you switch to/select a different mission, it will cancel the mission you are currently doing.
Other than that, there is no way to just cancel the mission and not start another (other than finishing it).
Best Answer
Based on experience in playing Tiny Tower, it is not possible to rename your tower.
I also did some research and it yielded no finds for renaming your tower.