You can enter each Form 1098 separately if you choose. To do this, you have to calculate the outstanding mortgage principal for each loan based on the average monthly statements, so in the case of a refinance with no cash out it is easier to just combine it on one form using the reported mortgage balance.
You are correct, if you refinance with a different lender, you can deduct the remainder of the points from the old loan. If it is with the same lender, the points are amortized over the life of the new loan. If you have points on your old loan that you have not fully deducted, and you refinance with a different lender, you should report each Form 1098 separately.
As far as Form 1098 being reported to the IRS, the only number that the IRS gets from your tax return is the deductible mortgage interest. All the calculations done to determine the deductible portion is done on worksheets that are not submitted to the IRS but kept for the taxpayer's records in case it is needed.
@DiddlyD
