I am looking for a way to create a form similar to one I found on a website

I put the website that has the form I want in your URL section of this form. Can something like this be don through your product?

Yes, this is possible in Jotform, but on a different layout. For the Date selection, you can use our DateTime field for this and limit the dates on a particular range of your choice.

https://www.jotform.com/help/337-How-to-Use-the-Date-Limits-Option 

For the time selection based on the dates, you do the same field type, a radio button field with the list of times as the options. Now, you have to conditionally show the time field based on the corresponding date selection. 

For show/hiding fields, follow our steps here: https://www.jotform.com/help/316-How-to-Show-or-Hide-Fields-Base-on-User-s-Answer. 

For the Images, you can use our Image Checkboxes widget (http://widgets.jotform.com/widget/image_checkboxes). To add the widget in the form, please follow the steps here: https://www.jotform.com/help/252-How-to-Add-a-Widget-to-your-Form.

The last set of fields are basic which you can find in the basic fields section of the form builder:

Here's a test form based on the sample form: https://www.jotform.com/70621187696969. Select March 9, 2017, to reveal the radio button for the time slot.

You can clone the form and finish it: https://www.jotform.com/help/42-How-to-Clone-an-Existing-Form-from-a-URL. 

Let us know if you need further assistance.