Are there companies that allow to be free when learning and teaching? I'm currently a trainer in a kind of "expert" domain: a web-programming backend framework.

However, what I truly like is to help people regardless of their level, mostly online or at conferences. The sad thing is that I have no time for this, because my employer makes a business of this, our clients pay us to train them and get more skills in this open-source tool.

Question:

I'm wondering whether there are companies in the world that just recruit people that will make learning &amp; teaching a full-time activity.

Let's take an imaginary example:

This company would have any kind of activity in the IT world, whatever business, and a few of its employees would spend time going to conferences to learn, or even give talks at conferences or in schools, and the rest of the time on Github, StackOverflow, public Slack, Gitter or Discord channels associated with this technology or in schools.

Is this even something that does exist, or is this pure utopia?
What you describe is pretty close to a university professorship. We are paid to learn and teach. We go to conferences and present our work there, as well as work with colleagues. We don't normally have strict no-compete clauses, though teaching for two different universities isn't normally allowed. Courses we develop can be used by others in many cases, online or otherwise. 

You would need to pick your field, however, since technology in itself is a pretty narrow subject. Study for a doctorate is normally quite theoretical in most fields, but there are programs, likely in engineering, that focus more on technology. 

Companies in industry have different rules. Normally they want to create a product and are focused on that. There are education-focused companies of course. However, most such companies have strict no-compete rules, some of them attempting to bind you for life. This is especially true if you learn proprietary material as part of your job. Also, the "normal" work day has expanded to the point where you likely just want to have a life in the time that you aren't "on the job". 

University professors, too, are always at work, but the work they are always at is normally thinking. 

Locally, you might want to visit Arts et Métiers ParisTech, and talk to one or more of the professors there.