As a trader, schedule C is only used to report expenses related to your trading activity. With a 475(f) election, the trading profit/loss is entered on form 4797. It does not flow to Schedule C but to Schedule 1 line 4. if you didn't properly make the election, trading activity is reported on Schedule D. Gains and losses from selling securities from being a trader aren't subject to self-employment tax.
I see no problem with your spouse being listed first on the 4797.
