Yes it does
If you have iTunes for Windows and have iOS apps saved on your Windows computer, you cannot actually use the apps in any way on the computer. The one exception would be if you jailbreak, you can be able to use anything on your iPhone's display on your computer. I am not sure how that works, but I know it is possible.
Sync your iPhone to iTunes on our computer
itunes will back up your apps when you sync your iPhone if the computer is authorized (store-authorize computer in itunes).
First download the apps onto your computer using iTunes. Plug in your iPod to your computer, open iTunes, and set the sync settings so that you can download apps to it, sync your iPod and you should see the apps on there. This requires an iTunes account.
Connect your iPhone/Pod to your computer/laptop, open iTunes and sync your iPhone/Pod. The apps will then appear in your "applications" tab of iTunes.
As long as your computer has itunes that is authorised on the same itunes account as your iphone, then it will automatically transfer all your iphone apps onto the computer
you cannot play iPhone apps on your computer. And if you want to open games from your iTunes then you cannot either. iTunes is only for songs or games, videos, photos and apps for iPhone.
You hook it up to a computer, go to itunes on internet, then buy apps.
Yes, via iTunes.
with the ipod plugged to your computer open itunes and go to File/transfer purchases even if the apps you have are free
check for available downloads or download them individually
with the ipod plugged, open itunes, go to File/transfer purchases to computer