1. There is a difference between http and not http because of the way the search engine parses the keywords you enter to search. Basically, "http://" contains special characters (: and /) which confuses the search engine. The colon is used for denoting special types of searches, such as link: allinurl: and site:. The slash has other uses of its own. When entering a domain to do a search, do not add http:// at the beginning. You will get wonky results (although sometimes the search engine will simply ask "Did you want to go to <link>?").

2. Don't enter http://. It won't do what you are expecting it to do. If you need to filter results based on protocol, use the appropriate option in the advanced search page. Do not enter the protocol into the search box.

3. http:// is not considered as part of the URL for search purposes. having a www. or not is important on the other hand, but that is a separate issue.

4. Sites without www will be indexed just fine. Again, it is possible for two versions of the same site to be indexed, one with and one without the www subdomain. This is called canonicalization, and someone wrote an article about preventing it, which is on this forum here: Canonicalization Prevention Guide

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