as long as I show that the money I got out of the refinancing has been
used for the property, then I can claim all of the interest and property
tax. Is that correct? Yes this is correct
Does paying the mortgage from that money count? Yes paying off the first mortgage to get a new one is fine.
Now if you take out more money and it is not used for the home -then you have a pro-rated amount that gets deducted.