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Isometry between two metric spaces implies Homeomorphism. But is it true that every homeomorphism is an isometry. I don't think this is true because $(0,1)$ is homeomorphic to $(0,2)$ but not isometric because the distances are not preserved. Let me know if I am right and correct if wrong.

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    Yes, it is far from true. To cite some intermediate notions: Isometry > bi-Lipschitz equivalence > uniform isomorphism > homeomorphism – 2017-02-02

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That depends on the exact formulation of your question.

  1. If both of your spaces have a given metric, then not every homeomorphism is an isometry as you already argued.

  2. But if you ask whether any homeomorphism can be realized as an isometry by choosing a proper metric on the target space(without changing the topology of it), then this can be achieved by just defining the metric on the target space by the pushforward of the metric on the source space by the homeomorphism.