-Will ads that WordPress.com posts on our free blogs be removed once I have self-hosting?
yes, but other ads might be put on instead, depending on the terms of your hosting
-Will mapping of my new domain that I have automatically happen once I set up the self-hosted plan with Godaddy?
no, you have to do it yourself
-Will I have more space than the 3gb already offered with WordPress.com
again, depends on the terms of your hosting agreement.
-Am I able to custom design or use the premuim themes at no extra costs?
yes to the first, no to the second. There are thousands of themes for Wp.org blogs, some free and some expensive. The premium themes here at WP.com are not available for WordPress.org.
-Will I be able to post my own ads with self-hosted plans.
again, depends on the hosting agreement. some places say sure, some say no.
-Am I able to put more than one blog on one self-hosted plan, or do I need a seperate plans for each blog.
again, depends on your hosting agreement. Probably you need to buy hosting separately for each blog.
-Am able to use the VideoPress option with a self-hosted plan at no extra cost?
No. Videopress is a WordPress.com thing.