Yes. Even if you are receiving Social Security benefits, if you have income from working at a W-2 job or from self-employment, you are still subject to paying in to Social Security and Medicare. Your tutoring income is self-employment income.
If you have self-employment income for which you will pay self-employment tax for Social Security and Medicare, you will need to use online Premium software or any version of the desktop software download so that you can prepare a Schedule C for your business expenses.
https://ttlc.intuit.com/community/self-employed/help/what-is-the-self-employment-tax/00/25922
Yes. You are a self-employed contractor doing online tutoring. You must pay income tax on the total amount you received from your 1099-NEC minus any expenses that you had in order to do the online tutoring. Which should include your internet costs as well as anything else that you needed to purchase. You will pay self-employment tax on the net amount as well.