If I am understanding you correctly, you can already get the sum of the amounts for your different sheets, all you need is the date limitation.
This is a relatively simple answer, use FILTER().
You'll have to change the way your summation works, but you can filter out, from each sheet, the exact numbers you need within a specific date range using FILTER. Here's what the formula looks like:
=FILTER(B:B, A:A>=G1, A:A<=G2, C:C="Widgets")
G1 is your starting date, and it needs to be inside of a cell with proper date formatting. G2 is your ending date, also needs to be inside of a cell with proper date formatting.
You can use any actual cell in your summary page to house the dates, I just used G1 and G2 in this case because it was available.
NOTE: You need to change the references to cells accordingly!
The above filter will provide you the entire second column (B) which you can then add together to give you the final number for that sheet.
=SUM(FILTER(B:B, A:A>=G1, A:A<=G2, C:C="Widgets"))
That will give it to you for a single sheet, you can just duplicate it to as many sheets as you need.
I added this formula to your spreadsheet
=ArrayFormula(if(row(B2:B) <= max(if(not(isblank(B2:B)), row(B2:B))),vlookup(row(B2:B),filter({row(B2:B),B2:B},len(B2:B)),2),))
It seems to deliver what you asked, except for the interpolation part.
Enter this formula in row 2, no need to drag down.
I hope this helps ?
Best Answer
Set the summation to cell
C5. If you insert a row above or below, then all formula's will be adjusted accordingly. You can add this to the cell as well: