Can I Enter Sleep Mode While Downloading File?

Solution 1:

Have a Mac in sleep mode while also downloading a file with iTunes, is currently not possible.

However the idea is not out the question, doing things while in sleep is possible on certain new Mac hardware running OS X 10.8, via the new Power Nap feature.

What you asking to do is currently not a feature of Power Nap, however it is something that might be possible with a hack or a future version of Power Nap.

Therefore Power Nap can only do the following:

When your compatible Mac goes to sleep in Mountain Lion, Power Nap still gets things done silently

• Mail. Receive new messages.

• Contacts. Your Contacts update with any changes you may have made on another device.

• Calendar. Receive new invitations and calendar updates.

• Reminders. Reminders updates with any changes you may have made on another device.

• Notes. Notes updates with any changes you may have made on another device.

• Documents in your iCloud account. iCloud pushes any edits you made to a document to your Mac notebook.

• Photo Stream. Your Photo Stream updates with new photos from your iPhone, iPad, or iPod touch.

• Mac App Store updates. Your Mac notebook can download updates from the Mac App Store.

• Time Machine backup. Your Mac notebook can back up while it sleeps.

• Find My Mac. Locate a lost Mac notebook even when it’s sleeping.

• VPN on demand. Corporate email updates securely.

• Configuration profile. Macs in managed environments can receive configuration profile updates.

Power Nap does more when your Mac is plugged in to an AC outlet

When your portable Mac is connected to a power source, it also downloads software updates, makes backups with Time Machine, performs Spotlight indexing, and can continue Mac App Store downloads. It also updates Help Center.

Solution 2:

If you put your computer to sleep all processing (including downloads) is stopped -> so no, you can't do that.

Putting the display to sleep (as mentioned in the question) is possible though.