Question 1 (that higher derivatives are not used) is yes.
Question 2 (getting decay without weights) is no.
Without weights, let $u$ be a compactly supported smooth function. Let $f_k(x) = u(x - k v) + u(x + kv)$ where $v$ is a unit vector. The family $f_k$ is uniformly bounded in any classical $H^s$ space. But the family $f_k$ is NOT uniformly decaying.