I've looked at the installations and the plugin you are using to store all this information. I will try to answer the best way I can.
From what I can see you need the User search function, because it looks like the doctors are actually the users on your site.
1. After looking into the options I have found that the “provider_city” user meta stores the city information about the user. However this is only a reference, meaning it does not store the actual name of the city, but just a reference number. Because of that the plugin does not see the actual city name, just the reference number in the database, which is not the infromation needed.
For this to work you will need a user meta field that stores the actual city name, in order to be connected with the user directly, and then enter the meta field into the ajax search pro user search section.
However returning specialties as results still won't be possible, because those are stored as user meta, and it's not returnable.
2. This is the same case as the 1.
3. To display the doctor just enable the User Search on the General options -> User search panel.
This would be much easier if the doctors were stored as a custom post type, and the information like cities and specialties as taxonomy terms (like categories). In that case, all of this would be possible, even returning the specialties.Best,
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