No, with domain mapping your blog would still be hosted here at wordpress.COM and all the terms of service and all the restrictions would still apply.
If you want affiliate links and advertising you are going to either have to self-host your blog on a web host or move to another blogging platform that allows advertising.
With self-hosting, you have to buy a domain name, hire a web host, install the wordpress.ORG software and then export your stuff from here and import it into the new blog. Any problems or troubleshooting with the installation, or later on, you will have to solve. Also, you will have to do all backups, upgrades and such yourself. If you are not afraid of the web and have at least a little experience with the backend goings on with websites and such, then you should be alright.
Self-hosting will cost anywhere from $7 per month to $50 or more depending on how much bandwidth and storage you need, and if you happen to get one of your posts "digged" then I've known people who have burnt through their entire months bandwidth allotment in a week or two and then they have to buy more bandwidth to get their blogs back online.
And unless you are getting over 5000 hits per month, don't expect to make more than the price of a scone and a designer coffee per month.