If you are talking about Q2A Core than I believe you don't have to worry about that. Gideon and other developer will take care of it..
However if you want to use it for your plugin or custom development, Than you should use system functions for databaswe but if you have to use mysql functions than I would suggest you to start to use PDO or mysqli so in futrue mysql get deprecated than you don't have to make related changes.. :)
Now mysqli or PDO it is a bit subjective and depend what you want to do and till what limit.. Here is nice article you can go through http://net.tutsplus.com/tutorials/php/pdo-vs-mysqli-which-should-you-use/
That's a good point. BTW you can change to mysqli pretty easily: all the functions are the same but with mysqli_* instead of mysql_*. Hopefully for 1.6 Gideon can fix that.
In the meantime: all the database code is in the qa-db-* files, so you can easily open those yourself and do said replacement.
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May 7, 2020: Q2A 1.8.4 released!