There's no universally agreed-upon definition of rude, and sexism and racism are very hard to define objectively. They're very context-specific and subjective, for the most part.
No, that's a double standard. But if we're using the modern redefined version of sexism, yes you're right.
First, being rude can mean pushing someone's buttons. People have easy buttons to push regarding their gender, orientation, race, etc. So even if your primary motivation wasn't to express hate for their whole category, you can contribute to * ism that way.
Second, a more frequent expression of rude behavior to people of a class that you don't like is *ism even if you aren't using the forbidden slurs normally associated with whatever prejudice you're expressing.
Sexism and racism aren't dependent upon another person's buttons being easy to push. That just means they're more sensitive than others.
Surely not, surely it's the reason for the rudeness - if the reason is based [at all] on the sex of the person one is being rude to then it can veritably be called sexism. Otherwise it's just being rude to another person.
There clearly isn't any easy way to measure "aggregate rudeness" to be able to prove that women receive a higher quantity of "just being rude", but that doesn't necessarily mean it isn't there.
I wasn't suggesting that it was a necessarily conscious thing either; just that it either is or isn't sexist based on other things than quantity. Aggregate rudeness won't even do, if you happen to see a woman you don't get on with every day and are rude to them then your aggregate rudeness towards women could be huge without you being sexist; equally you could hate and avoid women and so have a very low aggregate rudeness towards them.
It's like the people who get arrested while walking home from work because they happened to enter a protest area. The reason for their being there is totally unrelated to the protest yet the effect of their being there is to increase the size of the crowd by one.
We can argue until we're blue in the face that this is unjust and the truth is that it is: it's unjust for everyone involved. It creates further entrenchment and higher barriers between people.
But as many people have pointed out, quantity can have, well, quantitative reasons. I remember a blog post stating that being the only woman among 100 men at conferences, the author always got at least one stupid comment/question.
She even turned it into a formula:
#(women at a conference) = inversely proportional to #(times I talk about women in tech) [1]
Same with rude people. Let's say there's an even 1% of women and men who are rude to the opposite gender and they are equally prolific. If there are 10% women at the conference/company/..., a woman has a roughly 100x greater chance of encountering that type of behaviour. Even though the level of rudeness is exactly the same.
Approximately: 1100 people, 1000 males, 100 females. 1% rude makes 10 rude males per 100 females, roughly 10% chance; 1 rude female / 1000 males, 0.1% chance, 100x difference.
Considering that "badness" of rude behaviour is likely going to be on a bell curve, you not only get more bad behaviour, you also get worse behaviour. Just from he numbers.
So you can have outcomes that look subjectively sexist (women encounter much worse and/or much more bad behaviour) without there actually being any sexism.