Do I owe federal taxes because I received FAMLI benefits? The form I received for it was a 1099-misc from my employers insurance company.

Possibly.  Any portion of the insurance premiums that were paid by your employer, become taxable when the benefits are paid to you.  This means, if your employer paid all the premiums for the coverage, then it is totally taxable. 

If you split the premiums 50/50, then you would only pay taxes on half of the benefits your received.  When you enter this as a 1099-Misc, you will need to select that this did not involve work like your main job and did not involve an intent to make money.  Then if you did pay a portion of it, you would make a second entry to remove the amount that was related to your premiums paid.  Ex.  You paid 50% and you received $5,000, you would deduct $2,500 from your taxable income. 

Take the following steps to remove the income:

1. Income
2. Less Common Income
3. Miscellaneous Income
4. Other Reportable Income
5. Answer Yes to Any Other Taxable Income
6. Enter the description such as 1099-Misc received for nontaxable disability
7. Enter the amount on the 1099-Misc as a negative number. 

Yes. For the state benefits, yes, you would enter the amount in the 1099-G section for Unemployment and Paid Family Leave.