Only functions are methods.
The constructor chain appears to me to be a (lazy at that) short cut. On a side note, doing it the other way around completely would be acceptable solution that doesn't require internally constructing new objects. The constructor can certainly call any method it wants, so if you have an overwriting method, chain the constructor to that instead of the other way around.
Yeah, PDO and MySQLi are not compatible. Since you need to have MySQLi established to make use of a mysql_real_escape_string anyway, you should choose which to use. But more importantly properties should never be stored in an escaped format, regardless of if you intend to use a prepared statement bound variable or if you intend to use a simple string with escaped input data. Escaping is specific for the db, not for the data in question. It is completely legal for me to have a string with \' in the string, but by escaping it you have now corrupted it onto \\\' instead.