Greetings!
You ask some very good questions! I'll give you the answers within the best of my knowledge, though it never hurts to check the MLA (or APA) manual.
1) If the Red Cross's website does not have any links to other useful information you'll want to use a database like JSTOR or Questia; on Questia, I found several books about the Red Cross, including Champions of Charity: War and the Rise of the Red Cross, by John F. Hutchinson.
2) The main purpose of an in-text citation is simply to point readers to the correct entry on the "Works Cited" page. If there is no author and you are citing an article withing the website, put the name of the article in quotation marks within your parenthetical citation; if there is no title, use the name of the website and where to find it on the site, e.g., (American Red Cross-Home page). In your Works Cited list, list the article title alphabetically, as you would do with the author's name; if there's no title, or to cite an entire website using MLA :
Name of Site. Date of Posting/Revision. Name of institution/organization affiliated with the site (sometimes found in copyright statements). Date you accessed the site <electronic address>.
3), Yes, you're right. If you can specifically identify the section or page from which you are quoting, though, do so: ("Our History" par. 3). Any time you quote directly, you are supposed to give the page number, assuming there is one.
I hope this helps!
Thanks,
Sarah, EssayForum.com
Sarah, EssayForum.com