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It looks for Robots.txt wich doens't exist
robots.txt doesn't need to exist. You only need this file if you want to tell spiders not to spider some areas of your site.
Then request 404.shtml that also doesn't exist.
This isn't something Googlebot would look for in my experience unless you have referenced it somewhere (in .htaccess, for example).
Did you design your site? Are you sure there isn't a reference to 404.shtml somewhere?
Google has come about 2-3 times and so far hasn't followed a single link... I'm not sure why
This can happen ... Googlebot often grabs your index page first, then comes back for the rest of the site later. And it will occasionally just grab 1 or 2 pages.
A few more links into your site could help. It will give Googlebot more opportunities for finding you.