My question is does links to your homepage improve SERP rankings for your sub pages, since they are linked to from your homepage?
Yes it does but you should always link to the most relevant page and not reply on the user to find their own way there. If you have a really good article on your site link people straight to it.
Good question. I hope someone with lots more expertise that I have addresses... FWIW, I think I have noticed that subpages without backlinks of their own sometimes seem to be really hard to find... even if the home page that points to the subpage has PR of 4 or 5. Also, wouldn't subpages let you have content and backlink anchor text that more closely target a particular keyword?
Yes links to your homepage does help subpages to rank as well once the pagerank and link-juice reach the subpages, but the subpages will recieve less pr/juice then it would if the link was a direct link to the subpage.
Deep backlink = backlink pointing deep into your website. Eg individual blog pages rather than your homepage.