We claim that if $S=\{qrp_1,qrp_2,\dots,qrp_{2019},p_1p_2\dots p_{2019}\}$ such that $q>r>p_1>p_2>\dots>p_{2019}$ be prime number then $S$ satisfy all three conditions. The first condition is obvious. Let consider the second condition. Let $a\neq b$ be any elements of $S$. We have $$\gcd(a,b)=\begin{cases} \gcd(qrp_i,qrp_j)=qr & \text{which is not prime.} \\ \gcd(qrp_i,p_1p_2\dots p_{2019})=p_i & \text{which is prime.} \end{cases}$$All primes in $\{\gcd(a, b) : a, b \in S, a \neq b\}$ are $p_1,p_2,\dots ,p_{2019}$. Thus $S$ also satisfy the second condition. Let consider the third condition. Note that $qr\mid ab$ for every $a,b\in S,a<b$ since at least one of $a,b$ must be $qrp_i$. And so $qr\mid \sum\limits_{a,b\in A; a<b} ab$ which make $\sum\limits_{a,b\in A; a<b}ab$ to not be a prime power. $S$ satisfy the third condition.