Do I list 1099-G unemployment under wages & income and does the stimulus exemption automatically update?

Yes the Form 1099-G for unemployment compensation is entered in the Wages & Income section.  The TurboTax program will automatically calculate the exclusion if you are eligible. 

To enter unemployment compensation

Click on Federal Taxes (Personal using Home and Business)
Click on Wages and Income (Personal Income using Home and Business)
Click on I'll choose what I work on (if shown)
Under Unemployment
On Unemployment and Paid Family Leave, click the start or update button

Or enter unemployment compensation in the Search box located in the upper right of the program screen. Click on Jump to unemployment compensation

If you had taxes withheld from the unemployment compensation -

On the screen What's the total amount you received on your 1099-G? check the box Federal or state income tax was withheld on this payment.

The exclusion is reported on Schedule 1 Line 8 as a negative number.  The unemployment compensation received is on Line 7 of Schedule 1.  The result flows to Form 1040 Line 8.

Click on Tax Tools on the left side of the screen. Click on Tools. Click on View Tax Summary. Click on Preview my 1040 on the left side of the screen.