Not that I disagree with you about the importance of tire pressure, but a few PSI really isn't that critical for an average car. If you're racing in Formula 1 or Autocross, but for normal street driving there's a pretty wide margin, well within the accuracy of a cheap, working tire gauge. If the gauge is bent or sticking then you're better off getting a new one. Even expensive gauges can be off by a few PSI unless they're calibrated regularly, so having another gauge that you can use to check against is not a bad idea.
Like you mentioned, air temperature has a big effect on tire pressures, it's something like 1 psi for every 10 degrees. So even if the digital gauge can read 0.0001 psi accuracy, unless you're measuring the ambient temperature while you fill up your tires, , it's overkill. Also, while higher pressures do give better mileage, the increase is very, very minimal, on the order of a few percent, and probably not noticable unless you pay close attention to your mileage.
The sticker on the drivers door is a good starting point, but it's not always the optimal pressure. The car manufacturers put a priority on comfort rather than performance or tire wear. Plus those pressures might not apply to non-OEM tires. Sometimes there two different pressures listed, depending on the load. Generally, I like to inflate to about 32-36 psi for a regular passenger car radial tire, that gives the best compromise between comfort, handling, and tire wear. Above 38 psi the ride starts to get rough, and below 25-26 psi the car feels a little sluggish. Truck tires, low profile tires, heavy loads, and type of tire will change that number.