Yes, $A(X) = cX/\sqrt{\log(X)} + O(X/\log^{3/2}(X))$ for a positive real number $c$ which I think is 1. This result holds in much much more generality, e.g. for primes in congruence classes or even for positive-density subsets of primes defined by modular forms. See Theorem 2.8 of Serre's "Divisibilit'e de certaines fonctions arithmetiques" from 1976 in L'Enseignement Mathematique. You'd replace $\sqrt{\log(X)}$ with $\log^{1-\delta}(X)$ for a more general set of primes of density $\delta$, but the primes congruent to 1 mod 4 and 3 mod 4 respectively make up density $1/2$ subsets of primes. You can also get secondary, tertiary, or as many error terms as you wish.

Yes, $A(X) = cX/\sqrt{\log(X)} + O(X/\log^{3/2}(X))$ for a positive real number $c$ which I think is 1 (edit: This remark on the constant was just a vague recollection which is wrongly remembered as it turns out. See the comments below.). This result holds in much much more generality, e.g. for primes in congruence classes or even for positive-density subsets of primes defined by modular forms. See Theorem 2.8 of Serre's "Divisibilit'e de certaines fonctions arithmetiques" from 1976 in L'Enseignement Mathematique. You'd replace $\sqrt{\log(X)}$ with $\log^{1-\delta}(X)$ for a more general set of primes of density $\delta$, but the primes congruent to 1 mod 4 and 3 mod 4 respectively make up density $1/2$ subsets of primes. You can also get secondary, tertiary, or as many error terms as you wish.