1 Some 500 women across the country were asked if they agreed with various views once felt to be typically female.
2 Some 500 women across the country were asked if they agreed with various views once felt to be a typical female.
3 Some 500 women across the country were asked if they agreed with various views once felt being typically female.
4 Some 500 women across the country were asked if they agreed with various views once felt typically female.
“She felt like being typically female” seems unusual to me, and my guess is that it does not mean what you are thinking it does. It has a sense that she decided, on some particular occasion, that she wanted to behave in a typically female way.
Also note that #1 and #3 are the same, which I assume was not intended.