Can you choose a form template and then add customization?

I need to update a current sponsorship form that we created last year to collect sponsorship applications. There are a few fields that I would like to move around and add a few new ones. 

Yes, it is indeed possible to clone a form in your account and apply the needed changes to it, here's the guide that will help you to do it: How-to-clone-an-existing-form-from-your-account

Regarding to the image you shared, are you trying to get info from an external source and display it in your form? 

If so, then this is not possible, I mean, there is not an app or widget that will allow you to do it, but you can do it if you pre-populate the info in your form, this guide will help you with the pre-populate feature: Prepopulating-Fields-to-Your-JotForm-via-URL-Parameters

If you have any question, let us know.