In the "Help me figure this out" item, the box for amount spent on the original home, will be the principal pay off amount.
The balance on 31 Dec 2019 for the refinance loan is, as you surmise, the actual balance on that date or 1/1/2020 after the couple of payments.
Once you enter all 1098s and click "done" and then "wrap up tax breaks" you will come to a page that shows "itemized" or "standard deduction" chosen. Below that you should see a hyperlink that shows a "breakdown of itemized deductions."
In there you will see a reduced amount for deductible interest.
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