This has to do with the mass of things: every object, according to Einstein, increases its mass as it moves faster (that is, beyond a certain limit, the energy that pushes an object turns almost entirely in mass and only for an ever smaller fraction in speed).

This is a common error people make about special relativity. The mass of every single object is an intrinsic property of the object. It is constant. Period.

What varies is actually the energy of the object. The variable mass you mentioned is not the mass of the object, but is instead what is called its relativistic mass, or its energy divided by the speed of light squared.

The use of the relativistic mass should be avoided as much as possible, as it yields confusion.