= Solution
All assertions of independence below are relative consistency assertions. We construct structures satisfying the other <ZF> axioms but falsifying the indicated axiom; an ordinary model of <ZF> supplies the positive side. The convenient ground theory is <ZFC>, which has the same consistency strength as <ZF> by the <constructible universe theorem>.
For the <Axiom of union>, put $\lambda=\beth_\omega$ and use the <hereditarily locally small membership model>
$$
B_\lambda=\{x:\forall y\in\operatorname{TC}(\{x\})\ (|y|<\lambda)\}.
$$
It is a <transitive class>. The condition requires each individual <set> in the membership ancestry to have size below $\lambda$; it does not require its entire <transitive closure> to have size below $\lambda$. <Extensionality> and <foundation> are inherited. Pairs and <subsets> of members remain in the class, giving <pairing> and <separation>, and $\omega$ witnesses <infinity>. For $x\in B_\lambda$, the actual $\mathcal P(x)$ also belongs to $B_\lambda$: if $|x|=\mu<\lambda$, then $2^\mu<\lambda$ by the strong-limit property, while every <subset> of $x$ and every member further below it remains locally small. This verifies the full <power set> axiom. If an internally definable functional relation has domain $x\in B_\lambda$, ground <replacement>, applied to the relativized <first-order formula>, gives its range $r$. Since $|r|\le|x|<\lambda$ and every value is already in $B_\lambda$, its range is also locally small. Thus every <replacement> instance holds.
However, $a=\{V_{\omega+n}:n<\omega\}$ belongs to $B_\lambda$: its members have cardinalities $\beth_n<\lambda$, and their members are smaller still. Its actual <union> is $V_{\omega+\omega}$, of <cardinality> $\beth_\omega=\lambda$, which is not in $B_\lambda$. Transitivity makes any internal <union> witness equal to this actual <union>. Hence \b[<Union> fails while all the other <ZF> axioms hold.]
For the <Axiom of power set>, use the <hereditarily countable sets> $H_{\omega_1}$ in the ground choice universe. This is transitive and contains $\omega$. Pairs, <subsets> and <unions> have <countable> <transitive closure>. For <replacement>, a functional image of a <countable> domain is <countable>, and the <union> of countably many <countable> transitive closures is <countable>; hence that image is again hereditarily <countable>. These observations, with inherited <extensionality> and <foundation>, verify all the remaining axioms. Every real, viewed as a <subset> of $\omega$, belongs to $H_{\omega_1}$. A power-set witness for $\omega$ would therefore contain every such real, but a hereditarily <countable> <set> has only countably many members, whereas $\mathcal P(\omega)$ is <uncountable> by <Cantor's theorem>. \b[<Power set> fails.]
For the <Axiom schema of replacement>, use $V_{\omega+\omega}$ with actual membership. This limit-rank structure contains $\omega$ and is closed under pairs, <unions> and <power sets>: every relevant finite increase in rank stays below $\omega+\omega$. A separated <subset> has rank no higher than that of the original <set>. <Extensionality> and <foundation> are inherited. The internally definable <function>
$$
f:\omega\longrightarrow V_{\omega+\omega},\qquad f(n)=V_{\omega+n},
$$
is obtained by $n$ successive power-set operations starting with the parameter $V_\omega$. Each finite iteration, including its finite history, belongs to the structure. Its range has rank $\omega+\omega$ and is not an element of $V_{\omega+\omega}$. Thus the <replacement> instance for this <first-order formula> fails, although every other <ZF> axiom holds.
For the <axiom of extensionality>, take the well-founded cumulative <set> universe over two distinct <urelements>, and forget the predicate distinguishing <urelements> from <sets>. Each <urelement> has no members, just as the ordinary <empty set> has no members, so <extensionality> fails. Every other axiom survives in this pure membership language. <Pairing>, <union>, <separation> and <replacement> are witnessed by the corresponding ordinary <sets> of <urelements> and <sets>; <foundation> follows from their well-founded ranks, and the pure $\omega$ supplies <infinity>. Here <foundation> is written with “has a member” as its nonemptiness antecedent, and disjointness means having no common member. Formulations using inequality to one designated empty object are not equivalent after removing <extensionality>. There is also one necessary adjustment to <power set>: an <urelement> is vacuously a <subset> of every object because it has no members. Thus the power-set witness for $x$ is the ordinary <set> of all usual <subsets> of its extension, together with both <urelements>. This is still a <set>, and contains exactly the objects satisfying the unrestricted <subset> <first-order formula>. Consequently \b[the countermodel falsifies <Extensionality> alone], without silently weakening <Power set> to <sets> only.
Finally, the <hereditarily finite sets> $V_\omega$ satisfy <extensionality>, <foundation>, <pairing>, <union>, <power set> and <separation>. Every functional image of a finite domain is a <finite set> of hereditarily finite objects, hence also belongs to $V_\omega$, giving all <replacement> instances. No <finite set> contains $\varnothing$ and is closed under successor: it would then contain every finite von Neumann <ordinal>. \b[<Infinity> fails while the remaining axioms hold.]
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