How to make a cross-browser extension, preferably native, in 2018?

Question

C++ / C#

Is there a common API and what technology is currently used for this?

Answer 1

Regarding the second part of the question: It is
possible to write an extension in C++ and compile it to JS/WASM . Of course, you need to write it according to the available APIs in the browser. That is, interaction with many common C++ native modules will be unavailable, since there are no analogues for them in the browser.
For the first part, I will not give an exact answer, but:
Write under Chrome. In other browsers (Opera, Firefox) it will be possible to install via Install Chrome Extensions and Chrome Store Foxified, respectively.
In fact, Chrome and Opera work with extensions in the same way.