Say I post a blog on Tumblr or Blogger dot com with backlinks in my post, can I manually request google index those posts or my blog URL so Google sees them right away?
Yes ,you can go to search console and check the url once you recieve a green signal that page can be indexed .Whenever next crawling happens on the blog/ site following page is considered .Now this works in case of blogs on the same domains .Not aware of Tumblr or blogpost.com