Navigate webpage as if from my browser (Python, selenium)
Requests cannot process JavaScript, nor can it parse HTML and CSS to create a DOM. Requests is just a very nice abstraction around making HTTP requests to any server, but websites/browsers aren't the only things that use HTTP.
What you're looking for is a JavaScript engine along with an HTML and CSS parser so that it can create an actual DOM for the site and allow you to interact with it. Without these things, there'd be no way to tell what the DOM of the page would be, and so you wouldn't be able to click buttons on it and have the resulting JavaScript do what it should. 
So what you're looking for is a web browser. There's just no way around it. Anything that does those things, is, by definition, a web browser. 
To clarify from one of your comments, just because something has a GUI, that doesn't mean it isn't automatic. In fact, that's exactly what Selenium is for (i.e. automating the interactions with the GUI that is the web page). It's not meant to emulate user behavior exactly 1:1, and it's actually an abstraction around the WebDriver protocol, which is meant for writing automated tests. However, it does allow you to interact with the webpage in a way that approximates how a user would interact with it.
You may not want to see the GUI of the browser, but luckily, Chrome and Firefox have "headless" modes, and Selenium can control headless instances of those browsers. This would have the browser GUI be hidden while Selenium controls it, which sounds like what you're looking for.
I need to parse a page, keeping HTML and JS the same as in my own browser. Site must think, that I am logged using the same browser, I need to "press" some buttons using JS and find some elements.
When using requests library or selenium.webdriver.Firefox(), site think I am from a new browser. But I think selenium must help.