Sure, this has been studied, and various generalizations are known (infinite sets, three and more set summands, abelian groups other than the group of integers, asymptotic decompositions). You can check, say, the paper of Elsholtz and Harper Additive decompositions of sets with restricted prime factors and the references therein for a pretty comprehensive overview.

There seem to be more open questions and conjecture than results in this area. To mention just two, Ostmann's conjecture dating back to 1968 states that the set of prime numbers is asymptotically additively indecomposable, and Sarkozy has conjectured that the set of quadratic residues modulo a prime is indecomposible. (There is a nice argument of Shkredov showing that the set of quadratic residues cannot be represented as $A+A$, but it is not known whether it can be represented, say, as $A-A$; see this paper.) Two more links: Large sets in finite fields are sumsets by Alon, and Problem 4.11 here.

Sure, this has been studied, and various generalizations are known (infinite sets, three and more set summands, abelian groups other than the group of integers, asymptotic decompositions). You can check, say, the paper of Elsholtz and Harper Additive decompositions of sets with restricted prime factors and the references therein for a pretty comprehensive overview.

There seem to be more open questions and conjecture than results in this area. To mention just two, Ostmann's conjecture dating back to 1968 states that the set of prime numbers is asymptotically additively indecomposable, and Sarkozy has conjectured that the set of quadratic residues modulo a prime is indecomposible. (There is a nice argument of Shkredov showing that the set of quadratic residues cannot be represented as $A+A$, but it is not known whether it can be represented, say, as $A-A$; see this paper.) Two more links: Large sets in finite fields are sumsets by Alon, and Problem 4.11 here.

Sure, this has been studied, and various generalizations are known (infinite sets, three and more set summands, abelian groups other than the group of integers, asymptotic decompositions). Ostmann's conjecture dating back to 1968 states that the set of prime numbers is asymptotically additively indecomposable, and this is still open. You can check, say, the paper of Elsholtz and Harper Additive decompositions of sets with restricted prime factors and the references therein for a pretty comprehensive overview. An open conjecture of Sarkozy is that the set of quadratic residues modulo a prime is indecomposible. (There is a nice argument of Shkredov showing that the set of quadratic residues cannot be represented as $A+A$, but it is not known whether it can be represented, say, as $A-A$; see this paper.) Two more links: Large sets in finite fields are sumsets by Alon, and Problem 4.11 here.

Sure, this has been studied, and various generalizations are known (infinite sets, three and more set summands, abelian groups other than the group of integers, asymptotic decompositions). You can check, say, the paper of Elsholtz and Harper Additive decompositions of sets with restricted prime factors and the references therein for a pretty comprehensive overview.

There seem to be more open questions and conjecture than results in this area. To mention just two, Ostmann's conjecture dating back to 1968 states that the set of prime numbers is asymptotically additively indecomposable, and Sarkozy has conjectured that the set of quadratic residues modulo a prime is indecomposible. (There is a nice argument of Shkredov showing that the set of quadratic residues cannot be represented as $A+A$, but it is not known whether it can be represented, say, as $A-A$; see this paper.) Two more links: Large sets in finite fields are sumsets by Alon, and Problem 4.11 here.