Look, I understand that you have a stake in this. I get that. I am also male. I understand that you might want to look out for male interests in a sort of economic sense.
Nonetheless, what you are saying is deeply misguided and shows that you don't know what sexism became in the 20th century and looks like today. You are struggling too hard to make some sort of point. In doing this you are failing to come to a common ground and to understand the Other. I know this well; I have been there. I implore you to make one last effort.
Let us start from some common ground: women are granted a great deal of courtesy because they are in some sense 'overvalued'. The question you should ask is, how are they being overvalued, exactly? And the answer I think is, "they are being overvalued as romantic objects." It's a common thread with roots in that Shakespearean sort of ideal, the Quiet, Pure Woman who Knows Her Place and is Walled Away and must be Wooed by her Romeo. It plays forth in all of the behaviours you have insisted upon: that we would pay for dinner, buy them drinks, arrange the wedding, extend them romantic courtesies, forgive them their outbursts, et cetera. It is also the reason that they are discouraged from the club scene and thus can be whisked past the line.
This is our common ground. Now what you must understand is that this position is not merely a position of privilege, but is also deeply dehumanizing. Look at those words again: romantic "object". The same attitude which makes her precious also makes her little more than ornamental. And this is a story played out through most of our history, until several extremely noteworthy women in the Victorian Era decided "hey, screw that--we can be writers and mathematicians and factory workers as well." Of course, this soon after intersected with the Suffragettes, the World Wars and the Sexual Revolution. It was only very recently that women got the right to vote and the right to attend school -- much less the right to wear 'immodest' dress or the right to be a witch. For a long time they simply were not treated as human but rather as subhuman.
Sexism is simply that: it names these social institutions which treat women as a whole as subhuman. There are still many such institutions. Many of them come from this same Romantic Object Past that you are complaining about -- this is presumably why some man drunkenly licked her tattoo; he may well have thought that this might show that he was pursuing her and wooing her, in his drunken state -- in doing so he revealed that he doesn't really think of her as some independent person to be talked to, but rather some skin to be licked. They may also be inverted by the modern pornographic culture, as with the man who burst into a conversation asking for her to come to a pillow-fight essentially did the same, essentially saying "your conversation could not possibly be as important as our idolizing you as a sex object; come on, let's do this."
The bottom line is that the sexism isn't "in favor of" anybody, it's a deeply immature mistake that we've been carrying around from the Dark Ages to the present. It wouldn't be sexist if a female groped a man or invited him to have a pillow-fight because our institutions happen to not be geared to objectify men in this way. They objectify men in other ways though; especially in corporate culture where there is a tendency to emphasize those who neglect their families to work harder, to be cogs in a machine, to follow the orders of management. This may not be sexism -- that is, there is little reason to expect that women who join the corporate circuit are treated much differently -- but it certainly is objectification happening to men today. The pornography industry also objectifies men; in the industry men become little more than muscles, penises, and a bundle of perversions, with their stamina, girth, and virility being the most important qualities.
TL;DR = "rudeness" is not sexism, sexism is again the climate which generates, and is generated by, such rude acts. Key to the climate being "sexist" is that the climate lowers women beneath a baseline of what it is to be human. The examples you've given do not lower men beneath such a baseline, but there are cultural features which do and we should be wary towards those too, even though they tend to not be sexist per se.