Hi there,
I'm sorry you're having these problems getting your site setup and working the way you want it. Let me see if I can help get this all sorted out for you.
1. Are you referring to the VideoPress watermark that appears in the upper-right-hand corner of your videos? If so then I'm afraid that cannot be removed. If not, could you take a screenshot that shows what you are referring to and upload it to your media library? (http://en.support.wordpress.com/make-a-screenshot/). You can leave a link to the screenshot here and I'll be able to see it.
2 & 3. We have had several reports from users about recent pages/changes appearing and disappearing in the same way that you are. We are currently investigating what might be causing this problem. Sit tight and we will hopefully have a solution soon.
4. You can absolutely use your GoDaddy domain here at WordPress.com. I see you have now purchased the Domain Mapping Upgrade, and I can access your site through your custom domain. Do you still have any questions about this?
5. If you want to link to your external calendar from your menu, you can create a Custom Menu, and include that item as a Custom Link from https://theresapaints.wordpress.com/wp-admin/nav-menus.php. We have more information on creating Custom Menus at http://en.support.wordpress.com/menus/
Does that answer all of your questions? Is there anything else I can help you with?
Best!