$p$-adic series bounded if and only if it has finitely many zeros

(2) and (3) are equivalent. This is corollary 3.3 in Laurent Berger's IHP course notes Galois representations and $(\varphi, \Gamma)$-modules in 2010. In the same way, we can prove (1) and (3) are equivalent (in one direction, convergence is used).