***** NOT A TEACHER *****
1. "Double down" is a phrase that is being used a lot here in the United States.
a. One dictionary says that it means to increase your efforts or to focus even more on something.
i. My example: The police are doubling down on people who cross the street in the middle of the block. (That is, they are really, really going to look for people who do that and give them citations, or tickets. They will have to pay a big fine to the court. And maybe the police will even start warning people on TV not to "jaywalk.")
b. Therefore, in my opinion, that gentleman NOT ONLY said that ladies should refrain from laughing, but he ALSO went further: He claimed that there were no longer many "symbols of chastity." In other words, he double downed on the matter. (I understand that the phrase comes from the cardgame called Blackjack.)
2. "To NOT take something lying down" is easier to understand. It means that if you do not like something, then you are going to protest with words or even with action. In other words, you refuse to quietly and obediently accept something that displeases you.
a. In your example, it means that many young ladies in that country refused to keep silent after the gentleman made those remarks.
b. Here in the United States, for example, if your boss fires you (dismisses you from your job), some people do not take it lying down. They will go to court and tell the judge that the boss fired them because of their gender, religion, etc. (That is illegal in the United States.)