: 1) Would it be alright to simply move the blog site to www.blog.company.com (still on bluehost)? That would be fine if you do 301 redirects from the old URLs to the new URLs. If

1) Would it be alright to simply move the blog site to blog.company.com (still on bluehost)?

That would be fine if you do 301 redirects from the old URLs to the new URLs. If you do not do 301 redirects you are essentially starting over. The 301 redirect tells search engines, and your users, that those pages have moved and where they have moved to. This is important not only from a usability point of view but it also allows the links and PageRank you've earned to be properly assigned to the new pages. A small amount of value is lost but it's a whole lot better then nothing.

2) Should we transfer our blog site to be hosted on mediatemple?

That's up to you. There is no difference SEO wise between the two. The choice is whether you want to have two hosts or not. I would think one host would be simpler but you know your business better then I do.

3) Should we move our blog site to blog.company.com and get the blog index and compile it into redirection urls for the apache?

See my answer to #1

4) For readability reasons, we would rather have our blog site at company.com/blog.

There is no difference between subdomains and subdirectories as far as Google is concerned. So if you want to put your blog in a subdirectory go for it.