Yes, they do count but not as much as external quality links. I heard that it is useful to have at least one internal link on page, it interconnects the website better.
YES!! Internal linking is crucial. It helps keep your "link juice" in house, makes it easier for crawlers to index your website fully and frequently, and yes as already stated above they do count as links to your site.
Yes internal linking is very important to show the search engines which article on your site is the main article about 'keyword'. A good idea is to write one main article about your main keyword and three to five article about related ones and link them all to your main article and to each other. This way you can build an internal link wheel / net to each keyword you want to rank for and you have full control over this process, SY
6000 internal links is a lot for Google...wether it counts all the links depends on the linking scheme (how deep, how many layers...etc). Try to get a Google XML sitemap from Attracta SEO and you will definitely get all of your pages and links indexed.
probably, if a site has ONLY internal links then Google would not count them for much. However, if a site has both internal and external links in good quantities, the internal links definitely count
yes,google count all the internal links but the condition is that links should contain quality content and not be duplicate.
Internal links (more or less) help to say which part of your site is focused on what niche-related keyword. For example, it's generally accepted that you should link to a page targeting "Small Dog Training" from the main page of a site about "Dog Training".
It matters what you mean by counting. What it does is focus content to key pages. Google will use these pages for search results. It is not the same as a back link. If you have A 100 pages with similar content all linking to one page than google will assume that the page is important.