In $mathbf{ZF}$, it is possible for a set $A$ to be infinite but not to admit a countable set. In other words, for any $alphainomega$, there is an injection from $alpha$ into $A$, but there is no injection from $omega$ into $A$. If we replace $omega$ by a successor cardinal $kappa^+$ in the above statement, any $A$ of cardinality $kappa$ works, so there exists an example in $mathbf{ZFC}$. However, for limit cardinals, the question seems unclear to me.

Therefore: If $kappa>omega$ is a limit cardinal and $A$ is a set such that for any $alphainkappa$, there is an injection from $alpha$ into $A$, does it follow in $mathbf{ZF}$ that there is an injection from $kappa$ into $A$? Additionally, does the existence of such an $A$ imply the existence of a Dedekind-finite set?