Will using a hashtag in a relevant blog post's content that is shared with the hashtag on and indexed in a social network's search results for said hashtag give the blog post 'weight,' to search engine (Google) results for that hashtag?
Relevancy and user experience
No, however displaying or linking to hashtags such as Facebook, Twitter and Google Plus will as it add relevancy and user experience to your page. That's why its favorable to embed actual tweets on your content.
The understanding of what a hashtag is...
The hashtag craze was introduced to allow users to find and share relevant content for users, a hashtag that does nothing will dampen your user experience, Google may give some weight on trending hashtags but ultimately if your going to use hashtags then do it for the search engines and your users. Use hashtags to help your users find relevant content internally or externally. It's important to note that a hashtag can be used to find various relevant content on your own site without linking out.
Sharing content with hashtags
Often when sharing links on Facebook and Google it'll read the description of the page or og tags. This wouldn't include your hashtags, only that of the user input does, so sharing pages that has hashtags on them would not help at all, but reading the prior will.