Hi Amelia, you're right about that alt tags are for people who use screenreaders. Though search engines like google also use the alt tag to recognize there is an image file.
The alt tag is an onpage factor that influence ranking (http://moz.com/learn/seo/on-page-factors)
Although the alt tag may looks like as not important, it's worth using it.
About the spammy thing, using a word twice on a page has nothing to do with keywordstuffing 
Grtz, Leonie
