After e-filling wil Turbo Tax, do I need to mail my hard-copy W2s?

No, you do not need to mail in your W-2s. According to the IRS, 

"You don't need to send your Forms W-2 to the IRS and should keep them in a safe place with a copy of your tax return."

See the IRS site for more information: https://www.irs.gov/help-resources/tools-faqs/faqs-for-individuals/frequently-asked-tax-questions-an...