There is definitely something wrong with CPI. I'm not sure what it is because on paper it seems like a reasonable measure, but anecdotally my expenses are up more like 17% vs CPI's 7%. My "personal inflation" regularly beats out CPI (although usually not by such a high margin. Housing, fuel, and food have been the biggest increases for me. I wonder if all the consumer goods are somehow over represented.
It's a common misconception that gas prices barely effect those who don't drive. Unfortunately it couldn't be further from the truth. A massive portion of the logistics of America, bringing essentials that almost everyone consumes like food, healthcare goods, building materials for maintaining housing, etc are reliant upon vehicles that consume petroleum products to reach the consumer.
I don't think so, but I could be wrong. While gas does play a role in the price in a lot of different things like you state, the percentage of the final price of that item that includes fuel is usually very, very small.
Some back of the envelope estimates:
The last mile is usually going to be the largest expenditure of fuel per dollar of product transported. A semi truck gets 5 mpg on the low end and with diesel fuel currently at $5/gallon, that means it would cost just $2,700 in fuel to ship an entire truck load from New York to Los Angeles (much farther than most products would ever be trucked). Assume the truck is loaded with a low price, low margin product like toilet paper and you might ship around 4,000 small packs of toilet paper for a total fuel cost of $0.67 per roll. So if fuel prices doubled to $10 gallon, you would be looking at an increased cost of $0.67 per pack of toilet paper that is normally, maybe $10. So you are looking at a 7-10% increase in the cost of that toilet paper due to a doubling of fuel prices.
Of course, fuel is needed for the equipment that cuts the trees and the energy to manufacture that toilet paper, the plastic it is wrapped in, etc. But, we have to remember that we are talking about an inexpensive yet bulky product that we are shipping across the entire US and the weight is low so you could probably drag two trailers of it with only a marginal decrease in fuel efficiency.
The reality is that most products are trucked much shorter distances from ports that use enormous container ships where the fuel cost per dollar shipped is vanishingly small.
