Based on your inquiry, it seems likely you are using specific software to run your site. It is unclear what type of software is being used, which is a critical factor. It could be a CMS such as WordPress. a shopping cart such as ZenCart, a forum such as vBulletin, etc.
You would likely receive the fastest and most accurate response by using the support site of the specific software in use.
Based on your questions, you are in far over your head and should ideally step aside and find a programmer who can resolve the issue. With that said, I'll try to answer your questions.
"how would I go about determining exactly what content is missing from the site?"
In order to determine what is missing, you need a baseline. You need to understand the site's function and activity. For example, if you are running an ecommerce site, what is the timestamp of the last order placed on the site?
"What search engine queries would you perform?"
None. You have an onsite issue. That is where your attention needs to be focused.
** "Is there a plugin I could use to establish the inconsistencies between the database content and the search results?"**
No. The search results should not even be a consideration. Search engines may choose to index or not index your content based on numerous factors including the robots.txt file, the meta tags on each page, the content on each page and so forth. Asking this question indicates you are grasping at straws. If your site is important to you, hire a professional developer to fix the problem. If the site is not of great importance (i.e. it does not generate revenue) then you can visit the site of the software in use and spend a day or two reading various articles, forums and such, then making various setting changes in an attempt to restore the site.
Another option....contact the web host and request them to restore a full backup of the entire site. This option would likely be best, but you would lose all data from after the time the backup was taken.