Hello I am a licensed attorney here to help you with your question, please review my response and do not hesitate to ask for clarification.
In terms of the 1099 income you will have to declare the total amount as income, you do not pay any payroll taxes on it it would be considered as self employment income and since you do not have your own business, it will be incuded in your personal 1040 and your federal taxes would be paid at a higher rate.
The employer releasing themselves of tax liability could be a benefit for them and you as the 1099 income you will receive in a lump sum and even if you have to pay extra federal and state taxes on it, you would receive more since other withholding taxes are not comming out of the check.
If these were unpaid back wages, they should pay all the applicable taxes as now the burden shifts to you at the end of the year.
box 7 is only valid for nonemployees. Box 7 is the most common box to be filled out on a 1099-MISC. It is intended for independent contractors. If you are truly an employee - and not a contractor - and were paid with a 1099-MISC, you should complain to the employer and the IRS.
Box 3 is for "Other income." This box includes payments to you of prizes, awards or punitive damages. So this should be checked.
The clause included is fine, you should have the attorney who settled the case make sure the punitive damages they are paying is valid.