I can offer some answers to your specific questions based on my own experience from about a decade ago. Things may have changed somewhat since then, but probably not too much. (Also bear in mind that this is only one person's perspective and that requirements can vary from one university to another.)
I am not really sure If I should go into a master's studies or start pursuing a Ph.D. right after undergraduate school. Does getting a master’s degree improve the chances of being accepted into a Ph.D. program, or, perhaps, would it be a positive addition to my CV/ Resume for employment purposes?
In the US it's typical to go directly from a bachelor's degree program into a PhD program, with no intervening master's degree. The first couple years of a US PhD program are often pretty similar to a master's degree program: you take classes and do some amount of educational research, like a literature review rather than a truly novel research project. (Of course you can also be working with a professor on some original research on the side.) If you do get a master's degree and it is in the same subject (i.e. physics), you might be able to skip those first couple years, but this depends on the university.
- High undergrad GPA (3.8 out of 4 or higher)
Most graduate programs do not have a strict cutoff on what undergrad GPA you need to get admitted, and for those that do, it's certainly not going to be as high as 3.8 (except maybe at a few of the most exclusive schools, but even then I kind of doubt it). If you do have a GPA of 3.8 or higher, then that's fantastic, but even if it's as low as, say, 3.3, you'll probably still have a good chance of admission to many PhD programs in the US as long as the rest of your application is reasonably strong. The expectation here also depends on the reputation of your undergraduate university: if you go to a rigorous school which has a reputation for turning out well-prepared students, then you can get away with a lower GPA than if you go to a school whose standards are loose. I don't know offhand how Peter the Great St. Petersburg Polytechnic University fares in that respect, but based on some quick research it seems to be reasonably well regarded academically so that probably helps you.
Also note that your "major GPA" - that is, the average grade from among classes relevant to your major, i.e. physics, math, maybe computer science or engineering if you want to specialize in an area of research where those are relevant - is probably more important here than your overall GPA.
- High GRE score. Is it still relevant, though?
The general GRE (verbal and math) don't really matter. However, the subject GRE in physics does matter. It's similar to GPA in that there is probably not a strict cutoff, but a high score will be impressive, while a moderate score will not be a deal-breaker if the rest of your application is strong.
- Concise and meaningful application essay. Describe exactly what you want to do. Explain why you are applying there. Explain the specific details of the school that would contribute to your mission/goal positively.
I'm really not an expert on application essays but this all sounds pretty good. :-)
- Solid scientific publications record
This is not necessary at all. If you have publications, that's great and will certainly be a major benefit to your application, but as far as I know, the majority of incoming graduate students at US universities do not.
- Affinity for the school. Really know the college. Laboratories, professors, etc. At least be aware of them.
Again, not necessary.
Knowing a professor helps though. Specifically, if you've worked with or communicated with a professor at the university you're applying to, enough that the professor is willing to advocate for you to be admitted so you can work with them, then that will be extremely useful. This is one thing I wish I had known about graduate admissions when I went through the process: it goes better if you treat it more like applying to a job with a specific research group rather than applying to a university. But if you don't have that, no need to panic, it is certainly possible to get admitted without it and find a professor to work with after you enroll. (I did!)
- Letters of recommendation from known professors and people in the field would be a boost to the application
- Clean social media profile
This really doesn't matter at all. If there's anything extremely offensive on your social media then it might disqualify you, but it's also possible nobody will check.
At least, back when I got into grad school it would have been nearly unthinkable to take social media into account when considering applications to grad school. Maybe things are a bit stricter now.
I have also heard from a few interviews [...] that it is good to present your ideas for research and experiments in the application.
Yeah, probably true. Like many other things I've listed here, I don't think it's necessary, but it will help if your ideas are good.
One other thing I happened to notice while looking up information on your undergrad university: the Wikipedia article says
During the 2022 Russian invasion of Ukraine, the University issued a public statement strongly supporting Russian actions, calling the assault a "denazification operation" and accusing Ukrainian leadership of endangering the security and existence of Russia and "all of humanity". According to Andrey Rudskoy, head of the university, while scientific cooperation with Western universities had been developed over decades, after the invasion it was almost completely destroyed, with foreign universities halting their ties with the university.
Now, I don't know exactly what "halting ties" entails, but it's possible that US universities are declining to admit graduate students from Russia right now, and especially from Russian universities which have supported Russia's position in the war. If that's the case, then all your other preparation may be wasted. However, it's also possible that the political scene will look very different in four years; if the war ends and there are some significant changes in the position of the Russian government and Russian university leadership, then I can imaging US universities might relax any restrictions they may currently have. So this is not a reason to give up hope. I just wanted to point it out because it's an important factor to be aware of.