Erdős [Er84d] proved that \[H_k(n) \ll n^{2/3}\] (where the implied constant is absolute). The lower bound $H_k(n)\gg n^{1/2}$ follows from the fact that any set of size $n$ contains a Sidon set of size $\gg n^{1/2}$ (see [530]).

The answer is yes, and in fact \[H_k(n) \gg_k n^{2/3},\] proved by Alon and Erdős [AlEr85]. We sketch their proof as follows: take a random subset $A'\subset A$, including each $n\in A'$ with probability $\asymp n^{-1/3}$. The number of non-trivial additive quadruples in $A$ is $\ll n^2$ and hence only $\ll n^{2/3}$ non-trivial additive quadruples remain in $A'$. Since the size of the random subset is $\gg n^{2/3}$, all of the remaining non-trivial additive quadruples can be removed by removing at most $\lvert A'\rvert/2$ (choosing the constants suitably).