I have some non XOOPS pages on my hosting space (for a specific purpose) which I would like to show the XOOPS member links to registered users so can anyone please provide some help on how to query the XOOPS DB to achieve this.

I'm not sure I understand exactly what you want to do. I think all you're asking is query the database tables from MySQL. To do this you just need to connect to the database (you can find a tutorial here that walks you through connecting to the database. Then you just need to know the name of the table (which you can see in a tool like PHPMyAdmin) and then execute some MySQL queries (you can find the PHP MySQL functions here. Hope this helps start you in the right direction.

If you can give us a few more details we might be able to help more...

I have a folder like this outside XOOPS Path

FolderOutsideXoopsPath/index.php

On this page I want to show the XOOPS user menu
Edit Account | Inbox

I am not versed on MySql so my efforts have failed hence my quest for assistance.

Have you considered using something like either the pages or wraps modules instead? This allows you to "wrap" your page inside the XOOPS CMS which will then display your normal right/left blocks, etc. (including the User Menu). You can find the wraps module here.