State tax filing

1. No. The federal asks for your state unemployment payments.  State tax paid is a state issue. State and federal taxes must be paid quarterly by all employers. Taxes are paid in the quarter the income is earned. 

2. Do you mean the payroll tax for your employee, under their name and number? That is part of their state tax return. The money you withheld from your employee for taxes isn't reported. The Sch H is to report your share of the taxes, Medicare and Social Security.