Ticket fields based on drop down box
There is no way of changing the behaviour of the field based on the state of the ticket within the service desk. Clearly the functionality exists in the tool as, for example the resolution field and survey fields appear. However that level of customisation would require re-coding to achieve.
You will need to consider having multiple queues to achieve what you require. I would even suggest having a process and using the "convert to process" option if the additional information is required. The secondary ticket information could also be emailed to the user for the user to click n the link and complete the required data.