Sophia here from Shopify. Welcome to the Shopify Community, and congrats on launching your store!
I took a look at your store, and it seems that the .myshopify.com URL is still set as the primary domain. You can change this by setting a primary domain under Online Store > Domains in the admin.
The payment methods and checkout buttons are appearing as expected on the store. Customers can view the available payment methods in the footer of your website, and the Shop Pay and other accelerated checkout buttons will appear in the cart page when products have been added to the cart. If the cart is empty, it is expected behaviour that no checkout buttons will appear.
I haven't seen the payment page loop back to the first stage of the checkout before, but I'm happy to dig into this and find out what's going on. Please let me know which browsers and devices you've tried so far and whether this issue occurs across multiple browsers and devices.
In the meantime, you can also check any attempted payment history under Orders > Abandoned checkouts. If there was an error with the payment attempt, more detail may be captured in the admin. Did you see any pending payment authorizations on the credit cards that you tested?