Helpful, please post the sources about shaking hands from Reb Moshe.
In general, the way I look at it is like this. I personally didn’t create the terrible yetzer hara men have for women. In addition, I didn’t create women to begin with. If it were up to me, I would have created maybe only a few, and on a remote desert island. But they are all over the place. Many studies have shown, and Charlie who is an expert in Statistics can back me up on this, that up to 50% of all people are women! And this cuts across all socioeconomic, geographic and religious categories. Yes, every single religion and group has a huge problem of women. For a long time this was swept under the rug. Many groups would claim or believe that they were men only. Gradually it became known that this conforting thought was not the case. It was discovered that even among the most religious Jews, a huge number were women. This terrible problem of society in general which we thought we were immune to, managed to infiltrate our religion. Finally Rabbanim are dealing with this. And whether the families were rich or poor seemed to have no effect on the statistics. The very best families fared no better.
It is better that it is out in the open, because now we can come up with rational strategies for dealing with the female problem. It is nothing to be embarrassed or ashamed about if you have one in your own family. They are all over the street. It is likely that if you take a train, you will encounter these creatures. They have been known to go to stores, libraries, colleges–even synagogues! I don’t think anybody knows the true extent of the problem. The govt needs to invest huge amounts of money now to find a solution before it is too late. They are literally swarming about town, a silent infestation that threatens to take over.
But seriously, since there is no way to avoid them, it is hard to believe that the Torah wants us to give up every possible activity, because there may be a woman there. Are we really supposed to spend all our energy avoiding 50% of the population?
Note that in Sotah the gemara says that a husband who is overly strigent with his wife will cause embarrassment to her and people will say mai hai dka bodeles min haanashim. (What is with her that she separates from men.) There must be some happy medium which allows people to function normally, but stay away from potentially problematic situations. I don’t think that locking oneself in a room is the answer.