This is a great question and a pain point for sure that you cannot create different incident templates. If this were me I would do something like the below as a work around. Create a custom dropdown which would indicate the ‘issue area’ then create a workflow rule that would email the requestor with a form to provide additional information upon a new ticket creation which matches your issue area. This way you could specify different forms for them to complete. 

Regarding question 2 for the tech to attach the form to an existing ticket, I would create this in a canned response.

Another option is that you can create sections inside the incident form based on a custom dropdown. I did this for one client and it worked nicely. You would create a custom drop down in the incident template call it something like support area. Then select your listing of items needing support. Save the new incident form. Go back down to your newly created drop down and you’ll see a new button appear that says “add section” you could then insert “content” type field to describe the question and an answer field for the person to input the answer. The only thing I do not like about this is that the new field lives on the right pane when your ticket is created but this could be a viable option. 

You can do this through a service catalogue and create the form as a service item.

You can then create your own questions , work flows etc. It gets sent through this as a ticket.

You can then merge to an existing ticket if needed.