You should definitely adjust for inflation, as you should with all monetary variables. I wouldn't worry using the CPI, you could use an alternative indicator as a robustness check to see whether the choice of indicator affects your main conclusions.
CPI and PCE have already taken into account the "importance" of different goods and services based on how much an average household spend for items in those categories. You could look at the weights here.
Your intuition seems to be correct. Food and beverages take up 14%, Housing 42%, Transportation 16%, and Medical care 9%. These four main categories take ...
The PPI (Producer Price Index) measures the domestic output of raw goods and services.
The GDP deflator measures the changes in all goods and services produced in an economy.
For your purpose the US Bureau does release indexes that drill down further into the PPI in the form of industry level classifications and commodity classifications.
Am not a trained economist, but I would think this may be a temporary effect. A merchant may sell his newly purchased items for $2. Scruples aside, he could sell his existing inventory of that item (purchased at the pre-inflation wholesale price) at the post-inflation retail price. Again, an opportunistic, but temporary, advantage.
Edit: based on the video I get the logic that the person was trying to get across.
Basically the argument is that if in year $t$ deficit was 100mil but the same year the total debt 5billion and inflation was 10% that means that in real terms the value of debt is 500mill less and the teacher of that course treats this as 500mill revenue so under that ...