there are two different concepts in your first paragraph: the first it´s about people trowing their arms up in horror if a breast is shown on tv on a morning program, keeping the "drama" alive for at least a week by showing images of the guilty breast over and over again (its nipple concealed under a respectable black circle), attacking the shameless woman till her reputation drops as lower as it could be while showing a 16 year old model almost naked on the front cover of a men´s magazine for their visual sexual satisfaccion
it about people feeling uncomfortable and (some) disgusted when a woman near them in a bar e.g breastfeed her baby right then an there instead of hid herself in the toilet while women shown on tv commercial provocatively dressed (meaning: a barely there leather skirt and a shirt open all the way down to navel´s level showing two breast barely covered by a bra surely 3 sizes smaller than it should be) are considered a common thing to sell a new model car, a men cologne or a restaurant chain new offer of the week
that woman accusing you of belittling her quality as a person because of you being polite is just about her being an asshole probably in a bad day. the other telling you off for not being gentleman enough is probably one of those machist women that have contributed to prolongue stupid stereotypes through the years
sexism is prejudice or discrimination based on a person's sex or gender, and can affect any gender. So sexist is any action, attitude, idea.... that discriminates somebody because of her gender
feminists want women to have equal rights as men. If your girlfriend is more traditional, meaning she wants house keeping to be her priority and dedicate herself to raise your children, it´s her right (you may not want children or prefer the idea of both of you employed so you have a highest economic status. You´ll have to come to an agreement on that). What feminist don´t want is women considered to be the one staying at home by society in general, but that is bound to keep happening if the man earns 30% more than her having the same job, paternity time off is close to non existent and people is left to change their discriminating actitudes when they see fit.
Whether some people out there don´t want to accept the following comparison as valid, positive discriminating measures to achieve equal rights to women is exactly what were taken to achieve equal rights to black people back them. White people complain about their children having to adapt to black children´s academic level, having to share resources with them, having to share space on the bus ... summing it up they begin to complain about losing rights because of black people finally getting theirs. The end results had to be forced into equality because the ones used to have the privileged position would´t ever have renounce to it if equal rights were left to be given voluntary by white people
you should act following your values and forget you have to treat differently a woman because of being a woman, just consider a woman a person, consider a man a person, and if you are sure how you should react, treat and interact with a person, you´ll find it not difficult at all to know how to act