Why is there a consistent theory of continuum mechanics in which one just consider things like differential elements and apply Newtons laws? Is there a deeper reason for it. Is it the nature of newtonian framework that makes it happen or is it somehow related to nature of bodies (topological spaces with borel measure etc)?
What evidence do you have that there is a consistent theory of continuum mechanics? Certainly, when looked at through a macroscope, the universe looks like it behaves according to continuum mechanics, but this completely breaks down on the microscopic level. So you can't justify a consistent theory of continuum mechanics by using the universe. There's no reason that good approximations to our universe have to be consistent theories of physics; physicists currently believe that QED is a very accurate approximation to the electromagnetic force, but that it cannot be made consistent at the smallest scales without adding additional physics. And there are papers showing that if you just define your physics using, say, the wave equation without putting some kind of restrictions on the initial conditions, very funny things can happen. Also, Newton's laws of gravitation with point particles have some very unpleasant consequences. So I would say that, unless you're very clever about how you specify it, a theory of continuum mechanics would very likely not be consistent.
If you are very clever, you might be able to make a consistent theory of continuum mechanics, but I don't know of anybody who has actually done this.
I assume you're referring to condensed matter materials. The continuum limit is only an approximation which breaks down at the atomic/molecular scale. However, for materials many many orders of magnitude larger than the interatomic scale, we may use the continuum limit approximation if the system is in quasiequilibrium thermodynamically.
I guess you are talking about the Banach-Tarski paradox, which is not really an inconsistency in mathematics, it is a result following from the axiom of choice, I dont think only Newtons laws implies the negation of the axiom of choice, so I think Banach-Tarski vs mass conservation is resolved (ie can not appear), by the ways of which laws such as conservation of mass is applied to continuum mechanics.