There's so much cynicism from a subset of the HN crowd every time something interview-related like this is posted. Some of that cynicism is warranted, sure, but not in this case.
I'll keep it real: These aren't trick questions. They're simple, fair, and easy questions, and as such they're a good way to filter out people who have atrocious communication skills, or who possess mediocre theory of mind abilities, i.e. you can't put yourself in someone else's shoes and empathize with what they're looking for. If you find these questions particularly difficult, consider the strong possibility that you're lacking here.
"What's your biggest weakness?" This one is so simple it hurts: sincerely and honestly state a reasonable weakness. For example, "I've spent lots of time working solo, so while I've learned how to wear many hats, I don't have much experience working as part of a team." If you have terrible weaknesses that can't be overcome and would seriously impair your ability to do the job, you probably shouldn't be hired anyway. If you're completely unable to list any of your own shortcomings, then you lack self awareness.
The desire to lie and talk about a fake shortcoming often comes from a scarcity mindset: "I need this job so badly that I'll say anything they want to hear. I need to be perfect!" No, in most cases you don't. And if your interviewer is so unreasonable as to expect you to be flawless, is that someone you really want to work for? There are other jobs in the sea, keep looking.
"Tell me about a project you worked on." I couldn't agree more with what the person in the article said. Talk about YOUR contributions. The company is trying to hire YOU. What did YOU do? How did you do it, and why did it matter? Again, this is about communication. If you have a halfway decent theory of mind, it should be obvious what an employer would want to hear from you when you answer this question. Demonstrate your technical depth. Show that you're aware of and care about the impact your role has on the company's higher-level goals. That you can take initiative, drive things to completion, lead, execute quickly, handle complexity, learn over time, and make good decisions.
"Why do you want to leave your company?" People leave jobs. It's a fact of life. If you can't explain why you left, you seem clueless or even dodgy. Obviously if the job was still amazing for you, you'd still be there, so I can't agree with the article's advice to never mention anything negative. Just tell the truth, communicate it effectively, and address potential concerns. For example, if there were interpersonal issues, don't just dwell on those -- compare to other jobs you've had in the past where you've been able to work well with others together, and stress how important it is to you to work in an environment where people can communicate well, have good intentions, and support each other.
These questions just aren't that hard, and unless you have a seriously flawed track record, you don't have to be dishonest to give great answers. More importantly, those answers should be a reflection of YOURSELF. If a company doesn't like you, chances are you wouldn't like them. It's like dating. The goal is not to be a match with 100% of people. Just be your best self.
Corollary: If there's a test that you find particularly hard to pass in life, your knee-jerk reaction will always be to criticize the test. But it's important to consider your own potential shortcomings and ask, "Why is this test so hard for me?" That way lies self improvement.