Do I have to log into my JotForm account to reply to form submissions?

We are moving away from our current contact us form that is going into a messaging system because we would like to manage the form submissions directly through email.

With JotForm, I know it will send an autoresponder if I set one up, as well as a notification email. However, is the notification just to let us know to check the JotForm account or is it actually emailing us with the form information that the customer entered and then we are able to reply back to the customer directly?

If you set the "Reply-to Email" for your notification email to the email address field in your form, when you click reply from the email that is sent, the address the reply is sent to will be the address the user entered in the form. 

https://www.jotform.com/help/30-Explanation-of-Email-Notification-and-Autoresponder-Settings

This will allow you to reply to the users directly via email.  Form data and such will be included in the email.