The most recent CC 2014? The very latest? What's your OS?
Just installed an hour ago. 13.1.1
Additionally, my right click menus open to the left in 2014. How can I swap it to the right?
Disable tablet support in the Windows system control panel. This may in fact even bring back your Puppet tool, since I'm reasonably sure the position of the icons is simply calculated wrongly because of this setting.
Where, exactly, are you looking for the Puppet tool that you're not finding it? Show screenshots.
You do not download and install separate pieces of After Effects. Unless something terribly wrong and unusual happened with your installation, you have the Puppet tools, effect, et cetera.
Disable tablet support in the Windows system control panel
This brought back Puppet tool, thanks. What an odd interaction.
(Note, I did this fix right after completely reinstalling the 2014 CC apps, which I tried before and didn't work, so it could be fixed as a result of one or the other).