It is possible to change the name of the store. This can be done by going to Settings > Store details in your admin, however the change will not reflect in your store's URL.
It is not possible to change the Shopify username or the .myshopify.com URLas it is used for logging in, this is used as an identifier for your Shopify account. However, if you add a custom domainand make that your primary domain, the .myshopify URL will not be customer facing and visitors will only see your custom domain.
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