Eh, while you could argue it's unethical, he's saved Lois' life literally hundreds of times so I'm willing to cut him some slack.
That's exactly my point. Since these are fictional characters defined as heroes, we're willing to suspend disbelief/moral outrage about the things they do that would be deeply unethical or illegal in real life, like identity fraud, breaking and entering, assault, illegally imprisoning people in a particle accelerator, the nightmarish invasions of privacy Felicity Smoak commits every week, etc. In the real world, a lot of what they do would be criminal, but we choose to suspend those realistic standards for the sake of the story. So why can't we cut the fictional President Marsdin the same slack? She's done plenty of good herself, alien amnesty most of all.
Well it's a question, are Kara or Clark lying about the stories they're telling?
Yes, if they're pretending to be objective and unbiased reporters when talking about their own crimefighting. There's a clear conflict of interest there. They can't report both sides fairly when writing about a fight in which they personally took a side. Of course, everyone has some personal bias, but that's why it's essential to be honest about one's own stake in a story so that the public can be aware of one's potential bias. That's why news stories often contain disclaimers like "Full disclosure: The billionaire we're reporting on is a majority shareholder in the conglomerate that owns our newspaper/website." Part of honest journalism, part of earning the public's trust, is being scrupulously forthright about any direct or even indirect stake you have in the story, so that there's no appearance of impropriety or deception.
They showed the Golden Gate Bridge upon referencing National City. Does that mean It's located in the San Francisco Bay Area?
National City is essentially Los Angeles (where season 1 was filmed). What they showed was a montage of United States locations from roughly East to West: first New York City, then the Grand Canyon, then the Golden Gate Bridge, and finally National City (i.e. the LA skyline with "CatCo" digitally added to one of the buildings).
In principle, Clark laying about his identity and then leveraging that knowledge to gain employment isn't the same as what President Not Wonder Woman did. In a way it's worse and in a way, not so much.
For Clark, he took direct advantage of his secret to get his foot in the door. For the former President, there was no leveraging of that secret to gain an unfair advantage, only an omission of the full facts.
Yes, that's it exactly. They're both bad in their own way, yet we're willing to let Clark's actions slide for the sake of the story. It's not even really
about the relative immorality of the actions in the real world, because in fiction we forgive characters for all sorts of actions that would be criminal or morally reprehensible in real life. While it was right in-story for Marsdin to step down upon being exposed, that doesn't mean
we, as viewers, have to damn her as a person. She's not our president, so we have no personal stake in the matter beyond our feelings toward her as a fictional character and our investment in Supergirl's relationship with her. We can understand and forgive her actions, because we recognize that she's intrinsically a good person who made one big mistake.