When I make a new website, I often link to it from my blog. The link starts on the first page of my blog and goes deeper into the pages as I make more posts. Will the backlink lose value as it is no longer on the homepage of my blog?
That depends on the popularity of the page. If a page was popular it may show up on SERPs keeping the link working I would say.
The answer will almost always be yes, unless that particular post keeps getting links pointing directly at it, rather than the blog home page.