This one was not easy to solve ...
Error: The crawler could not communicate with the server. Check that the server is available and that the firewall access is configured.
Context: Some pages were not crawled (giving the above error) and, what is worse, all the sub content of that site was not crawled either! (the pages were the homepages of the site)
Solution:The pages that could not be crawled due to this error contained a custom web part. This web part used default credentials for a given action. During crawling, the SP_Search account tried to perform this action but did not have the appropriate rights. This gave an error that stopped the crawling for the whole site.
This blog helped me: http://patricklamber.blogspot.com/2010/04/why-might-moss-crawler-not-working.html