*cracks his knuckles*
BRING ON THE QUOTES!
Frankly, this just sounds like lechery to me. At best it is unprofessional. If your job is to check tickets, then check the tickets. By all means converse with the passengers, but if all you can say is stuff about their appearance or imply that they're out to get laid, you should shut up.
Being nice and friendly is fine. Unsolicited comments about someone's appearance however are generally low-level sexual abuse. Now, I'm not saying that you can't compliment people on their appearance, but you should not do it unsolicited or with people who are not your friends. When you are friends with someone you can judge what level of sexual talk is appropriate between yourselves. Perhaps you both want to share bawdy jokes, perhaps you'd rather just keep things entirely professional. When you know someone, you can judge what they will find appropriate. If you don't know someone, you cannot judge what they will find appropriate, and so you should not attempt to discuss their appearance at all. Please note that by solicited I mean either they your opinion is explicitly requested, you are in a situation where such behaviour is expected, or you have been given implicit permission as signalled through body language.
It's all contextual. If they were dance partners then they'd have got to know each other pretty well, and from reading each other they could work out what was and was not appropriate for them.
Some people are just plain idiots. I consider holding a door open just generally a nice thing to do, the only time I'd not do it is if I know the person behind is a jerk.
Friendships are started over shared interests. "You look pretty" is not an interest, it's a comment that you find that person sexually attractive, and thus are implying you would like to have sex with them. The power dynamic is to make the person in your debt because you gave them something. It's inappropriate in most settings, and especially in a professional context, where the passenger is already in the official's power. If you must try and make friends with passengers, a better comment might be "you're reading (book) - I've heard about that, would you recommend it?"
That's kind of the point. A lot of sexism is trivial. It's not people going "nice baps", it's more borderline stuff, which is just mildly skeevy. The hundreds of unsolicited compliments women receive on their looks (I, a man, have probably received less than 10 unsolicited compliments from people I don't know, and all were about the quite safe topic of what my t-shirt had on it). The way women's ideas get marginalised, the way they get interrupted more than men, the way their experience counts for less, the fact their life experiences are not considered credible if a man has not also experienced it. That is what most sexism is.
No doubt that a lot of people embellish their stories. That's human memory, it's pretty ****. So sure, consider the details of any individual story suspect. But then ask yourself, why would so many people lie, in such a consistent way? If ten people see a red Ford go by with its headlights on, and you asked them, you'd overall get a picture of a red Ford with headlights on, but you would get random memory drift - someone would say it was blue, another would say it was a Vauxhall, another would say it had its indicators on - but you'd probably get 7 people saying it was red, 6 people saying it was a Ford, 8 people saying its lights were on. But that's not what we have here. Here, we have thousands of people telling remarkably consistent stories. The details may vary, but if you have that many people telling you something, you should probably believe it.
You don't arrest someone just because one person claimed they did something. But we're not talking about punishment by law of individual people here, we're not saying "Joe Bloggs wolf-whistled Victoria Pennythwaite on the tube", we're saying "women on the tube are wolf-whistled by men". Whether you can prove any individual person perpetrated an act is not the point, you can certainly however prove that such things do happen. It's sort of like a firing squad - the reason there's a squad is so that (in principle) you can't tell who actually killed the man, it's a shared culpability.
I just want to state at this point that there is individual sexism, and there is systemic sexism. Individual sexism might be denying someone a job because of their gender, or being the subject of unsolicited sexual comments. Systemic sexism is that less people of one gender apply for that job in the first place because it's not considered "manly", or there are physical requirements which make it harder for women to make the grade, or where for some other reason there is a skew in results by gender. Men rarely suffer from individual sexism, but they do regularly suffer from systemic sexism. Now, in the following I deal with generalities, so there will be exceptions, but overall the following holds:
I doubt that when shooting at British troops, the enemy considers much whether that trooper is male or female, so there being more male combat deaths is not in of itself sexist. But the fact that more people in the military are men probably is to a degree sexist, though in a more systemic way. Societal expectations of boys tend more towards physicality, so more boys than girls take sports and fitness seriously. Similarly, societal expectations of boys are to be more aggressive than girls, and so they are more likely to think "I want to join the army". Finally, women are physically less imposing than men (in the median case), and thus fewer women meet the physical requirements than do men.
That men make up the majority of suicides is institutional sexism, it's the whole "manliness" thing gone awry. Men feel obliged to be lone protectors, to be "strong", and if they can't manage that, and do not possess the emotional wherewithal to get help, then they may think of themselves as a failure and thus that their life is not worth living anymore. We do condition boys to believe that losing is weakness. Women meanwhile are conditioned more to form support groups, and thus if they break down they have people who can help.
That men make up the majority of workplace deaths is again institutional sexism. Men tend to be the ones who do more dangerous jobs, and also the ones more likely to suffer in silence rather than seek help.
Regarding men losing in family courts, again this is sexism, there is an institutional bias here towards women because women are seen more as the carers and men are more likely to be the main financial provider (a downside of women not having equal pay, but also because women tend to strive for a better work-life balance). In all sorts of family areas, men lose out - we see less of our children, we likely get less time off for paternity leave, we're considered the main breadwinner. It's not fair, and it is a problem. No one would deny it's a problem. And it's why men should be feminists. Because the point of feminism is not that women are better, it's that they are equal, and we should not have gendered expectations of what a man should do and what a woman should do. Men can stay home to look after the children, women can be the main breadwinner. Women can be assertive and intellectual, men can be nurturing and artistic. Boys can play with dolls, girls can enjoy Meccano. This is the society we want - where everyone can be the person they want to be, without being shamed because it's not what their gender is expected to do. Where people are judged based on their merits, not on their looks or their genitals. Where if you do the same job as someone else, to the same skill as that person, you get the same pay.
Equality is not hard to understand, but to get there we need to question our entire society because for so long it has been built around powerful men doing what they want to and stuff the rest of us. We don't see the inequality because it's what we think of as "just the way things are". But is it the way things have to be? Is there a way we can be better?
It will be difficult. You will have to make sacrifices. Your behaviours are sexist, they are racist, they are homophobic. You will have to change. I am sexist, I am racist, I am homophobic. I will change. Because at the end of it, I will be a better person. And so will everyone else. And then I will be able to treat others equitably, and be treated as such in return.
TLDR: Yes, the guard in the OP was creepy and should go on a sexual harrassment training course.