If we start with axioms of $\sf ZF$ replace the axiom of Infinity with an axiom stating the existence of a set that doesn't have an injection to a von Neumann natural, and replace the axiom of Power sets by the axiom of unbounded cardinality, that is for each set $x$ there is a set $y$ such that $x$ is injective to $y$ but $y$ not injective to $x$.

Is it consistent with this theory to have all its sets being Dedekind finite?

Can this theory interpret $\sf ZFC$, through constructing $L$ within it?

If we start with axioms of $\sf ZF$ replace the axiom of Infinity with an axiom stating the existence of a set that doesn't have an injection to a von Neumann natural, and replace the axiom of Power sets by the axiom of unbounded cardinality, that is for each set $x$ there is a set $y$ such that $x$ is injective to $y$ but $y$ not injective to $x$.

Is it consistent with this theory to have all its sets being Dedekind finite?

Can this theory interpret $\sf ZFC$, through interpreting $L$ within it?