How do I input construction material that I get reimbursed for, as a COGS, or an expense account? If COGS how do I bill them for it, if expense what tax line do I use?

I would use an expense account I create called materials expense if it were me. Report it under other expenses.  And the **reimbursement** is not reimbursement (only employees are reimbursed by the company for something the employee paid for), it is income to the company.  On the P&L income is reduced by expense, so if those two values are the same the net would be zero effect.

COGS on the income tax form has a section where you enter the inventory starting and ending values and inventory purchases, the math on the form to get COGS for taxes.  So I like to keep my QB COGS account only used for inventory sales.