Good Morning Dick,
I've used IR thermometers for years in the HVAC business, and because I use it at home, I can offer my opinions. First, your not in NASCAR, so your not looking for the same detail they are. A few degrees difference in temperature across the tread, or from tire to tire is perfectly normal.
To answer your questions:
1) No. I/R's are not really needed, but they are relatively cheap and knowledge is power. A good tire pressure gauge is sufficient normally. Set your pressures cold and check daily while traveling.
2) Accuracy is dependent what you but you buy, of course. The one I own is about 1%. That sounds pretty good, but the range of the device is roughly 500 degrees, so 1% gives about a 5 degree accuracy, not so good in the big picture. I think you are more concerned with repeatability though, because your looking for the change in temperature across the tire, from tire to tire, or over time during operation.
3) I/R's are simply a non-contact thermometer. They have a laser for aiming, and have conical focus area that they measure temperature through infrared detection. The farther away from the surface, the larger the focal spot gets and your measurement becomes more of an average over that area.
4) The I/R will give you the surface temp only. What would the temperature of the air tell you anyway?
5) The I/R will give you the temperature of the SURFACE of what you shoot. The bearing is inside a cup, covered in grease, etc. If you shoot your wheels near their centers after driving for an hour, and find one wheel is 100 degrees higher than the other 3 wheels, you should be very suspicious of that wheel.
6) For normal use around the house and RV, I wouldn't get too carried away. I'd be leery of the $19 device, $50 to $100 seems reasonable. Consider specs, but again don't get to hung up on accuracy. If your device is off by 2 degrees, it will generally still be off by 2 degrees an hour down the road too. Reflectivity of the surface, like a shiny wheel, will throw the I/R off a bunch so you might have to take several readings and approximate them anyway.
7) 50 miles from nowhere and your tire is heating up? Let it cool down, check the tire pressures, slow down, get to a service center and have it checked.
8) How often do you suggest we check the tires? (I guess the answer would be...as often as you want or, as often as you stop) Exactly. Very hot conditions, rougher roads, loaded heavier, and higher speeds all cause increased tire heat. I'd check more often under tougher conditions.
The long and short, if you got a hundred bucks to spend, an I/R and a GOOD pressure gauge are both good tools. A good pressure gauge probably being the more critical of the two. The I/R will certainly come in handy for many other applications (Oven, A/C, furnace, refrigerator, etc.) and provides you with more info to make good decisions so can be a good investment too.
Hope this helps.