You can update the store name in Settings/General - What you place there is what your theme pulls from, so once it is updated, you should see the new name reflected in the footer of your website.
Do you have the domain added to your Shopify admin in Online Store/Domains? If yes, did you purchase the domain through Shopify or did you purchase it from a third party?
I suspect that either the domain is being spelt incorrectly, or the renewal may have lapsed and may be leading to a placeholder site until it is either renewed or purchased once the renewal period ends. In either case, we'll want to take a look at the account to determine why it is happening.
While we’re not able to provide account-specific support via the Shopify Community at this time, we’d be happy to continue assisting you through live chat, email, or callback. Please visit https://bit.ly/3cJkx8V and log in to your account to create a support request.
If the ".myshopify.com" URL has previously been claimed, it can not be used by any other store, even if the store is inactive. You will need to create a store using a similar name such as "domain123" to create your store and receive your free ".myshopify.com" URL. If you own the custom domain you wish to use for your business then you can add this domain once the store is created. Once added, this will replace the domain that is seen publicly when customers view your website.