As far I know, Shopify does not charge for trials. It offers a FREE trial of 14 days. Sorry if I didn't get your question. And you can contact Shopify support, they will definitely resolve your issue.
Welcome to Shopify, delighted to decided you set up a store.
Can I just confirm if you have purchased a plan yet at all? If you start a trial and purchase a plan in the free trial, the trial will still run it's course for the 14 days. But even during the trial and definitley when you pick a plan, you can make your store publicly active if you like. You would need to remove the password protection (when you're ready to do so) and then it should be available. You can remove the password protection by taking the following steps:
From your Shopify admin, go to Online Store > Preferences.
Go to the Password protection area.
Uncheck Enable password.
If you paid for a plan but didn't receive any confirmation, then that is a little odd. We can't look into specific account details in the community right now but it could be worth contacting our support team directly via live chat or email here and confirm this with one of our support advisors once you've verified your details.
Are there any other questions you have about Shopify, setting up your store, specific features you might be looking to add? If so, don't hesitate to let me know!