Looking for module to do the following...

There's no single module for this-- this is essentially what drupal does. To emulate this basic functionality, at minimum, you'd need core drupal, http://drupal.org/project/cck, and http://drupal.org/project/views. I would probably also include http://drupal.org/project/imagecache and http://drupal.org/project/imagefield cck field module as well for the pics.

Using your specific example, you would create a content type (basically a page type) for "greyhound", add cck fields for the individual pieces of information (sex, birth, weight, color, etc.), then use views to display a list of the greyhound pages. You could have a field for "availability" that would indicate whether or not the animal was 'available' for adoption and use views to only list greyhounds that were still available (this would require someone to edit the page and update the "availability" field once the animal was adopted).

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I would use the Content Construction Kit (CCK) and views modules plus some theming. (you may want to use Drupal 5, not sure of the modules current status in 6).

I would start by making a content type that represents a greyhound (or whatever), use the body for the description and one field for each of the details (including the image). I would then theme the output for the content to look as desired. I would then use views to make a view that shows content of my new type.