Or maybe unemployed and day trading (investing) and if so, what information would you enter on the application for employment?
This question is better directed to any potential lender, the question is not legal in nature. There is no legal barrier to a creditor lending you money if you are unemployed, receiving unemployment insurance, or quasi self employed. So long as you can meet the lenders minimum requirements and qualify, then they can lend you. However, candidly, it would be very difficult for someone only receiving unemployment to get any meaningful credit. A mortgage is almost certainly out of the question (no consistent income history).
You reference an application for employment. Also that the one you reference is on unemployment, and participating in day trading. Without a steady source of income, most lenders will deny the extension of credit. It is not clear what the circumstances are, and what you are asking.