Yes, you can each contribute to an FSA, as long as you don't plan on getting married next year. (If you do get married, the excess amount will be disallowed and added back to your taxable income, but there is no penalty.)
You would each need to claim one child as a dependent and pay for care for that child, and you will need separate receipts from the care provider for each child.
Only one person can file HOH, that would be the person who pays more than half the total household expenses. (It must be more than half, but even $1 more is enough. If you contributed exactly half each, no one could claim HOH.) The other person will file as single with the other dependent.
