You have got several questions there. Before answering, it is important to clear few misconceptions that I assumed from your wording and sample code. There are 3 types of Shopify Apps.

1. Public
2. Custom
3. Private

So if you are building a Private app, then provided code will not work for creating Product because Private apps use Basic authentication while Public and Custom apps manage authentication with OAuth 2.0.

I'm using Isomorphic Fetch in my project, which should work server and client side.

Even though it works on Server and Client ends, do not call Shopify API from Client side as this will expose your private app credentials.

To implement what you are trying to do, you need to modify React App as well as Backend code.

1. Add an event listener to Button
2. Send a POST request to Backend server
3. Create Product on Shopify via API
4. Add the product to cart using ID returned in previous step
5. Return the Cart URL in response

Sample code in React would look like

function Product() {
  const addProduct = async () => {
    const cart = await fetch("/your-end-point", {
      method: "post",
      body: JSON.stringify({
        // add product params
      }),
    });
  };

  return <button onClick={addProduct}>Add Product</button>;
}

ReactDOM.render(<Product />, document.getElementById("root"));

Then inside your Node.js application, handle the Shopify API part. Instead of using fetch, I will recommend using Official Node.js module for Shopify.

Sample code will look like

const Shopify = require("shopify-api-node");
router.post("/your-end-point", async (req, res) => {
  try {
    const shopify = new Shopify({
      shopName: "your-shop-name",
      apiKey: "your-api-key",
      password: "your-app-password",
    });
    const product = await shopify.product.create(req.body);
    // use product ID from previous request
    const checkout = await shopify.checkout.create({
      /*checkout params */
    });
    res.status(200).send(checkout);
  } catch (ex) {
    console.log(ex);
  }
});