Let's say I wrap up a game after it's done. I own the code, I own the ip and half of the art. The other half of the art comes from free sources: they are either in the public domain or under a free commercial use license like creative commons. The game is distributed in a single self contained package.

How do I go about registering a copyright on this final package? Can I put a disclaimer saying the copyright applies only to my original work in the package? Do I have to cherry pick all my original content out of the package and put the copyright on that?

You talk to a lawyer because you've created a bit of a mess.

In the mean time while you wait to talk to a lawyer, keep documentation of where you got every single asset. Keep documentation of the license you acquired it under, the website and URLs or people you obtained it from, the original author listed in the web site, and copies of the license.

It is sadly common for people to get content under one license and then re-release it under a different license. Many photographs and designs that are licensed end up getting re-distributed where someone claims they are freely available. Having a documentation chain that shows you obtained the file from a specific source under a specific license can help you avoid costly problems later. If the file was owned by someone else and unlawfully redistributed the license trail shows you still acted in good faith, then you get to negotiate new terms with the proper owner.

Be wary of sources that claim to be put into the public domain. Some "anti-copyright" statements are not legally binding. Others correctly release economic and property rights but fail to release rights like moral rights for attribution, integrity, or author's reputation, and more. While the person releasing the content is unlikely to sue, there are legal rights that are tricky to revoke. They can be an issue if someone wants to buy/license your game outright or buy your company.

If you don't have documentation that you have the proper rights, you're asking for trouble if you use them. Keep them documented.

As for the act of registering a copyright, that part is easy. Go to copyright.gov, fill out some forms, include the first 25 and last 25 pages (usually that means the file with main() and another random file), and pay the $35 fee.  It's $55 if certain other conditions apply.  Wait a few months, get a certificate.  File the certificate in a drawer somewhere. Your copyright is registered. 

Enforcing your registered copyright is much more difficult. That requires going through the courts, which is expensive and time consuming, winning the battle or negotiating a settlement, and extracting money which often requires collections services.

9 minutes ago, frob said:

In the mean time while you wait to talk to a lawyer, keep documentation of where you got every single asset. Keep documentation of the license you acquired it under, the website and URLs or people you obtained it from, the original author listed in the web site, and copies of the license.

I do so in the credits section of my game. If I got only one specific asset from a source, I will have an entry such as:

"Picture of a dog"
by Bobby Watson
(freeart.org/bobbywatson)
Licensed under Creative Commons 3.0
https://creativecommons.org/licenses/by/3.0/

If I got a bunch of different assets from a same person, I will either name each of the assets or I will put a title that represents what those assets are followed by all the credentials like the example above.

22 minutes ago, frob said:

Enforcing your registered copyright is much more difficult. That requires going through the courts, which is expensive and time consuming, winning the battle or negotiating a settlement, and extracting money which often requires collections services.

I want copyright as a mean to defend myself in the first place. If someone takes my content and attacks me claiming it's his own, I will have proof that I didn't steal anything.

1 minute ago, Michael Aganier said:

I want copyright as a mean to defend myself in the first place. If someone takes my content and attacks me claiming it's his own, I will have proof that I didn't steal anything.

Registering your own copyright won't help with that.

For that concern you need to talk with an actual lawyer, and do research on things like legal clearance, legal risk tolerance, and documentation trails that are legally acceptable. Everybody's situation is different. Laws vary by location so advice you get on the Internet probably has differences with the law in your spot on the globe. 

Those are why you need an actual human lawyer who understands your actual situation and your actual laws, your actual needs and concerns.  Advice from strangers on the internet, even lawyers on the internet, don't know all those details without in-depth discussions.

27 minutes ago, frob said:

Be wary of sources that claim to be put into the public domain. Some "anti-copyright" statements are not legally binding.

This is a nightmare. Some that are legally binding isn't recognized by all countries. Where the content is from is very important.

The main problem you will find is that sometimes what was marked as "Public Domain" is content uploaded without the original creators knowledge. Weeks after you publish the game you get a call from someone accusing you of theft and removing your game from stores.

18 hours ago, Michael Aganier said:

I own the ip and half of the art.

Wouldn't it just be best to get copyrights for your own content, instead of the game as a whole?