I inherited a web system that I need to develop further. The system seems to be created by someone who read two chapters of a PHP tutorial and thought he could code...
The situation isn't as bad as you think it is, unless you already have lots of non-Roman characters (that is, characters that aren't representable in Latin-1) in your database already. Latin-1 is a proper subset of utf8. Your web app works in utf8 and your tables' contents are in utf8 as well. So there's no need to convert the tables.
So, try changing the SET NAMES latin1 to SET NAMES utf8. It will probably solve your problem, by allowing your php program's connection to work with the same character set as the code on either end of the connection.
Read this. http://dev.mysql.com/doc/refman/5.7/en/charset-connection.html