Objectification can be of anyone, not just women, but it appears to be worse off for women.
I think the most useful way to explain this is to highlight the difference between the object and the subject.
Is the woman the subject of the sentences narrating her life (or just this particular interaction), or is she an object in the sentences? Do they do, or are they done to?
This doesn't always work (for example, "That person works for me."), but it's a good gut check.
Disney princess movies are simple and good for contrasting examples.
In Sleeping Beauty, Aurora was born, and a witch was jealous of her and cursed her. Her parents and fairy godmothers hid her. The witch still managed to trick her and put a sleeping spell on her. A prince found her, kissed her, and rescued her, so she married him.
Aurora never really did anything. She never drove the plot; the plot happened to her.
Anyone else could have been switched in for her character and not changed the outcome. What makes her an individual didn't matter.
In Mulan, Mulan was prepared by her family to meet the matchmaker, but the matchmaker was not satisfied with her. Mulan snuck into the army in order to save her father. She struggled initially in training but ultimately succeeded in saving the emperor from the Mongols. When Shang came to her home, she invited him to dinner.
Something happened to her, and then she reacted and did impressive things.
Mulan does not objectify the main character because the movie is so focused on what she does. She takes an active role in everything past the matchmaker (and even the disaster of that interaction was driven by her unique characteristics). Some other character with different values and emotions would have dramatically changed the story.
Being the subject of sentences is important because it generally means that your thoughts, wants, needs, and actions are considered.
"I'd hit that" is objectifying because the hittee's subjectivity is not considered, and the dehumanizing tone of "that" discourages thinking of the hittee as someone with actions, thoughts, emotions, and preferences.
The sexual interest is not the problem. The lack of assumed agency is.
Wanting to do something to someone is objectifying. Wanting someone to do your bidding is objectifying. But it's not objectifying to want someone if you care about their individual subjective characteristics, their wants, needs, and emotions. It's a confirmation of someone's internal subjectivity to want to make them happy.
The question details ask about complimenting someone on their looks.
Complimenting a person's looks doesn't generally draw attention to what they do. It praises their function as someone who is looked at.
When someone has done very impressive things, like Kamala Harris becoming California's attorney general, calling her "the best looking attorney general in the country" highlights her importance as an object of critical visual evaluation.
However, when someone is going out and has spent an hour on hair and makeup, complimenting their appearance is also a compliment on their preparations, so an appearance-related compliment is more appropriate in a club than in the office, where most people hope that competence is their defining characteristic.
The question details also ask about objectification of people like bank tellers, who are treated as important only for what they can do for you.
Classism and the commodification of human interaction are issues. Treating someone as a robotic fulfiller of your needs and wants is objectifying.
But if you're polite and approach the interaction with some empathy, then you're treating them as a person with an internal emotional world that could be affected by rudeness, not as a robot.
Now I'll address sexual objectification, which the OP also asks about.
I think it's discussed a lot because it's so inescapable in advertising and other media. People are constantly exposed to shameless, unapologetic, undeniable sexual objectification and have a lot of opportunities to form strong opinions about it.
The Society Pages published an excellent article, Sexual Objectification (Part 1): What is It?, and I'm going to go through the test questions they came up with and expand upon the subject/object distinction I discussed earlier.
Does the image only show part of a sexualized person's body?
Will this backside do things, or will things be done to it?
Does the image present a sexualized person as a stand-in for an object?
Does a thing (like a table) want, feel, or do?
Does the image show a sexualized person as interchangeable?
If people are interchangeable, their individual wants, needs, and emotions are erased. Fully realized people have so many variances in preferences and actions that they cannot be interchanged.
Does the image affirm the idea of violating the bodily integrity of a sexualized person who can't consent?
Clearly, things are being done to a person in something like this situation; she is not doing things.
Does the image suggest that sexual availability is the defining characteristic of the person?
The presentation of indiscriminate sexual receptivity eliminates the concept of the person's preferences. Such a person does not chose but is chosen.
Does the image show a sexualized person as a commodity (something that can be bought and sold)?
Again, such a person does not chose but is chosen and will, presumably, robotically fulfill the wishes of the purchaser.
Does the image treat a sexualized person's body as a canvas?
Canvases are written on and then looked at. They don't actively do anything.
The reason that people make a big deal about it is that normalization of sexual objectifiction of women leads to rape culture, which has been discussed a lot recently because of the Steubenville High School rape case and other high-profile cases.
If those boys had been conditioned to always think of women as people who do things (including actively enjoying sex) and not people whom things are done to, it's less likely that they would have taken advantage of an unconscious female. They would have thought more about her subjective experiences rather than what they could do to her body.
I believe that people act out on bad ideas they were taught and are not inherently bad. Problematic media themes plant the seeds for social problems. What happened to Daisy Coleman is a worst-case scenario that plays out if boys are taught by a perponderance of media messages that women are doees, not doers.