As far as I know you can't do what you are requesting using Forms or Forms Pro alone will get the job done for your ticketing needs as it can't do the custom URL to your specs nor will the search be easily done in my opinion.
I would suggest going a step further and using Microsoft Flow to capture the request details and having Flow then capture and process the details.
As an example, I've set up a simple internal Service Desk system using Forms to capture details (Forms can do branching questions which is great to capture scenarios) and then having Flow look up the submitter's Office 365 profile details and use aspects of those details to then populate an email and send it to the Service Desk inbox for action.
A future enhancement will be to save each ticket to SharePoint and update it accordingly.
The email can be customised with any info you want.