NP, NP-complete and Reducibility

Question

Suppose there is polynomial time reduce from problem $A$ to $B$.

fact $1)$ if problem $B$ is $NP$-hard then Problem $A$ is $NP$-complete.

fact $2)$ if problem $A$ is $NP$-complete then Problem $B$ is $NP$-complete.

fact $3)$ problem $A$ as not hard as $B$.

Can I say fact $(3)$ is true and fact $(1)$, $(2)$ is false because:

the fact $1$ is false because if problem $B$ is $NP$-hard then $A$ can be $P$, or $NP$ or $NP$-complete?

the fact $2$ is false because if problem $A$ is $NP$-complete then problem $B$ can be $NP$-complete or $NP$-hard?

Is my reasoning correct?

Answer 1

For 1), let both $A$ and $B$ be halting problem. Then $A$ is clearly reducible to $B$ and $B$ is $NP$-hard (any computable problem can be reduced to halting problem in polynomial time), but $A$ isn't $NP$-complete (as it's not in $NP$).

For 2), let $A$ be any $NP$-complete problem (for example, 3-SAT) and let $B$ be halting problem. Then again $A$ is reducible to $B$, but $B$ isn't $NP$-complete.

For 3) - it's unclear what "problem $A$ as not hard as $B$" means.