Apps that are purchased on my devices are deleted from the devices when I sync them with my desktop. iTunes states that the purchases are being transfered, but they are still deleted and do not show up under the iTunes app list. What is causing this?
For some reason when it syncs, it sometimes only remembers the iTunes information and doesn't read the device information. So you may find that after syncing it will deleted apps you'd installed and install apps you'd deleted. It's very annoying. To get around that, I have set my devices to not automatically sync when plugged in. When I do plug them in, I right click the device name, click Transfer Purchases, and after that has been completed then I sync.
There's not enough information to diagnose what's wrong with your iTunes and/or iPhone/iPad.
That's definitely a bug, since transfer of apps into iTunes should happen, did you seek customer support?
PS: Make sure your computer is authorized to have those apps. It's under "Store" menu -> "Authorize This Computer". Enter same Apple ID and password as you do when buying apps on your iPhone/iPad.