Is there ever a time when a Qatari lady is ever held legally responsible for herself? After the age of 18 or 21 say?, when she can do all things without some males permission or counter signature?

The answer is no, you cannot get a passport at anytime in your life without a male relatives permission.
As you are a Qatari prehaps you would be honest and enlighten us as to what you cant do without male permission? - do you need a males permission to recieve hospital treatment as in Saudi? I would hope not as I do believe nowhere could be backward as Saudi.

"Deaths in the Bible. God - 2,270,365
not including the victims of Noah's flood, Sodom and Gomorrah, or the
many plagues, famines, fiery serpents, etc because no specific numbers
were given. Satan - 10."