Credit-Card Authorization (via Authorize.Net) scenarios

I am using Authorize.Net as an integrated Payment Method in one of my Forms.

I am trying to ascertain how JotForm (and perhaps Authorize.Net) react in the following situations:

Successful Payment Submission:

This appears to issue the Thank You Response... inclusive of some acknowledgement of the Payment information.

Of course, there is recognition of the Payment in Authorize.Net... and acknowledgments are sent to the Customer and Merchant.

Incorrect-Incomplete Payment Submittal Information:

This seems to "immediately" reject (without tons of explanation) and return you to the Payment-Widget on the Form.

If not corrected, the Form will not Submit... which is fine.

Can anything more be accomplished for this situation?

Correct-Complete Payment Information (but the Payment is Denied):

What happens in this scenario?  I haven't been able to test this situation.

Is JotForm or the Customer properly notified?  How is it evident in JotForm?

Does anything get written back to the Submissions file?

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Thank You in advance for your clarifications.

Your explanations are right for the first situation. In the second situation, the form will show an error message and tells to the users to fix the issue. 
Generally, our users experiencing the issue when they forgot their integration in the sandbox mode. The real card credentials are not working when the payment field in the sandbox mode. 
Guide: How-to-Test-Authorize-net-Payment-Tool-on-Sandbox-Mode 
And when a payment is completed, it means you received a submission. We are unable to track the next steps. The issue may be located on the payment gateway in this scenario.