Hello SirMax.
Removing the "/page/" is easy enough with Mod Rewriting or I'm sure some plugin or another but the issue will have more to do with the weight the post is being given by the engines than the URL as the final URL is the same. To illustrate, there's an article on Search Engine Watch at http://searchenginewatch.com/sew/how-to/2179376/internal-linking-promote-keyword-clusters that outlines how the math of internal links work (simplified). Essentially what's going on is that your homepage is the strongest page (at least I assume you're part of the 99.99% of sites like that) and even in a blog, the blog homepage will be the strongest page within the blog, so pages it links to get the most weight. When one of your posts moves onto page 2 it's not the URL that causes the drop it's that the link to that post is now on a weaker page.
The related posts plugin is a solid idea but won't fix the specific issue you're having (though it will help in other ways). What you probably want is something like this plugin that displays the most popular posts. What you can do is place a widget on your homepage and set the posts to a reasonable number (10 or 20 max) and it'll display always your most popular posts on the homepage (read: blog homepage) helping insure their internal link weight is maintained. This will weaken slightly the actual post links as they stand today by adding more links to the page but it should help you address the issue you're having.
One concern to be aware of here is that you do need to review these pages. Some posts are popular simply by age and won't give the highest value for current traffic. You'll want to occasionally review which pages are appearing and if there are posts with low current traffic that are blocking other posts (those just below the post number threshold you set in the plugin) then you could be losing traffic so you'll want to exclude them.
Hope that helps. 
