The error you found in the log file has nothing to do with the current issue: if you did not change the personalization settings in web.config the most probable cause for this behavior would be the authentication cookie expiration. For example, you visit a page, wait for a long period (so the cookie could expire) and than try to perform a personalization task (putting the part on a page, changing the part property, or something similar). MonoX is telling you than you don't have rights to do it since you are not authenticated anymore. Let me know if this doesn't sounds right.
Going back to the initial issue, if you plan to remove Login.aspx you should be aware of some consequences - for example, it is set up as a default redirect page for forms authentication (search for login.aspx in the web.config).
The behavior you are describing (blog part appearing on one page and not on another) can be easily understood if you take a look at how personalization works. Every time you personalize a page via admin toolbar, it saves the current settings in the database, with the currently active URL as a lookup key. In your case, you did some changes (added a blog part) on the root domain ("/") and saved them to the personalization system. However, "Default.aspx" is considered as a different URL by the personalization system, as you may change the default page name at any time, so it renderes the default content (without the blog part). Our recommendation is to use "/" in the navigation menu (by adding an "external page" with the "/" as the URL in the page management) to avoid such situations and maximize SEO rating for your site.