For the questions/issues that are related to the blog component exclusively, let’s move them to the Landing Pages Blog topic to keep this one focused on the plugin only. In any case, for what I can see the problem it’s mainly that the theme (which is very basic) doesn’t fit your use case, and seeing that you already have updated it looks like it’s more on your side to keep adjusting it to your specific needs (the default should be “good enough” after the update). I might put some additional time on it, but as mentioned it has been unmaintained for a while and it’s not really my main focus.
On your initial issue, the name already exists error usually happens either when recreating a page manually or when resetting the remote repository while keeping the pages. Either way the solution is just to remove the existing pages and syncing again.
On your issue with the “Test Connection”, the last plugin update now takes both the URL and the branch for the validation so make sure the data you input is correct (assuming that your fork is https://github.com/nomadtown/blog-landing-pages I have no issues testing it with the master branch).