So let's imagine a scenario. It's spring, and it's a fine day for a ride. Go in the back of the garage and dig out the bike from all the stuff piled on top of it. Shake off the dust off we go for a ride. What a beautiful day it is. The ride is nice but we are soon tired. Maybe we should have exercised more over the winter... But the ride on the bike sure was cushy.
Dismount the bike and admire it for its beauty, it's so good to have it. Check the saddle, pedals, brakes (after the ride), the chain could use a little oil. Check the tires next - wow, these tires sure could use some air.
So we get out our tire pressure gauge and check the pressure. 30 psi! Wow, it sure lost a lot of air. Aren't we lucky we didn't get a pinch flat? That was close. Ok, where's the pump - oh, here it is. The sidewall on the bicycle tire says 50 psi, and when last checked they were filled to 50 psi - last year. Okay, now that we aired the tires up some, let's see what a difference it makes.
Standing to the left of the bike, grab the handlebars, put the kickstand up and away (because every Good Bike has a kickstand) while tilting the bike left swing the right leg up and over the saddle - ok, so far so good. Position the right pedal near the top of it's stroke, but a little forward, set your foot on it, and when the coast is clear stand up on that right pedal and...
Whoa! What a difference a little air makes.
While it's a comparison with a lot of limitations and differences, it serves as a simple example of how air pressure can affect rolling resistance. With less volume, the tires on a bicycle are even more affected by air pressure.
There are many reasons why a vehicle manufacturer will post inflation criteria on the door jamb sticker. If you are uncomfortable using higher pressure than what the manufacturer says to use, then by all means don't inflate higher than that.
But if you want to improve fuel economy, try increasing the air pressure in your tires. But only to your comfort level. Doing so will certainly reduce the contact patch with the road - absolutely. But how much? I think (key word there) this difference is negligible. And increasing air pressure does not change how the tire's tread compound grips the road. The surface of the tire where it meets the road remains the same, there's just less of it and the tire flexes less and in turn reduces rolling resistance when inflated with higher pressures.
But you are right - it's not for everyone. But it's wrong to think that reducing air pressure in tires will somehow improve fuel economy.
I'm curious - how often do folks use their roof racks? Is it used every day, maybe a couple times a week or month?
An internal combustion engine is designed to operate near the boiling point of water. The sooner it gets there, the better fuel economy will be. I'm using an engine block heater year round (if I remember to plug it in) to help get up to operating temperature sooner as my commute is about 15 minutes through suburban city routes. You could monitor your coolant temperature using one of several available OBD-II devices. However, you shouldn't need something like that just yet - take care of a few other things first.
For example, let's go into your typical drive. It would be good to cover speed, number of stops, terrain, and others.
And sorry to hear about your problems. Here's to better days ahead.