Are MySQL queries executed/indexed by search engine bots?
Yes and no. Any queries that you execute while displaying a page are executed when the bot loads the page too. Bots can not, however, execute queries on their own, neither do they invoke any Ajax or other javascript.
Think of the bot as a user with Javascript/Flash/other dynamic capabilities disabled.
Search engine bots will only fetch pages using the HTTP GET method. So, if a GET of one of your pages results in a MySQL query, then that query will get executed when the bot requests the page.
(Note that this is why you should never have a link resulting in a GET request that does something destructive, like delete a page or data from your database. Instead use a form that specifies the POST method.)
Unless you have a really big security breach in your system, the answer is no. It would be a really bad idea to have a search engine be able to index your queries because this would reveal the internals of your databases to the greater world.