I'll just tackle the 182.
"Full" power is whatever the engine will produce at full throttle with the mixture set appropriately for the available air density. Both how much fuel enters and the ratio of fuel to air is controlled by the carb and the mixture control.
The engine in a 182 is working against the propellor which is biting into the air. Since it's a "constant speed" prop, without going into how those work, the propellor is going to attempt to hold a constant RPM by adjusting how large an angle of attack the blades are biting into the air with. Too slow, the blades go flatter and take less bite allowing the engine to speed up, too fast they'll take a bigger bite and slow the engine down by adding more drag.
So if the propellor is always trying to stay at a particular speed, we need some other way to measure how hard the engine is working. For most piston airplanes this will be the manifold pressure gauge. It's literally a measurement of how hard the engine is sucking, but again without getting into it too deeply, it's a measurement of how hard the engine is working.
Many piston singles with constant speed props have a manifold pressure limit. Especially those with turbos that can keep feeding air well above the manufacturers continuous rated power setting. The 182 has no such beast, and it's rated to full sea level air density power at full mixture rich (as much fuel as we can give it per the carb settings) and as much air as it can suck in to burn it.
As you've said, the manufacturer wants to see at least 125F rich of peak. In other words, more fuel than it will burn at the most efficient peak fuel/air ratio. Some of the fuel carries away heat from places they don't want overheating. Some of the fuel is there to prevent detonation at high pressures which will destroy the engine.
So, now you take off. You start to climb. The air gets less dense. What happens? The fuel air mixture is now even more rich, too much fuel not enough air entering the engine. You lean it back to 125F rich of peak and keep climbing. Let's pretend you never touched the propellor control (the O470 as installed in the 182 has no time limit on max RPM which is where the propellor governor will hold the engine speed with the prop lever full forward, but in practice everyone will bring it back a bit for noise abatement).
You keep climbing. What happens to manifold pressure? The engine now has less fuel and less air to work with. Manifold pressure falls. The engine is no longer operating at peak power.
Up here in Denver, my engine isn't producing full rated power right at takeoff. Especially on a hot day. Density altitude is the control for how much air is being sucked into the engine for burning and the air is thin at a 5885' MSL airport on a 90F day.
The charts in the POH tell the tale. Notice how the top manifold pressure number and the top RPM number fall off as you gain altitude. In cruise at 10,000' MSL or higher, you're not producing much power at all. The prop blades have been set to a number but they're not necessarily taking as big an angle of attack bite at the air as they did at your sea level takeoff. The engine simply can't produce enough power to take that big of a bite and still maintain the RPM requested by the prop control, so the blades move flatter and take less of a chunk of air at each revolution.
I can't tell ya what the POH or AFM says for the PA-28-200R. If it's not in there, sometimes you have to dig through the engine manufacturer's stuff.
An example is the "green arc" on the 182 RPM gauge. There's literally no explanation of it in the Comtinental powered 182s. It's not a prop limitation, it's not an engine limitation, and it's not a combined vibration or other limitation. It's just marked. Probably because the range is what the POH has power numbers for. It stops before max RPM but there's no limitation that says there needs to be a gap there, and there's no limitation other than noise and engine longevity that says you can't fly it around all day at max RPM. It's a made up marking. Literally.
The green arc on the MP gauge is mentioned only as in relation to heat. Operation outside of the green arc on MP on an O470 may not produce enough heat to keep the carb warm since its mounted on the bottom of the engine, and carb ice may form.
Of course anyone who knows anything about carb ice knows that MP isn't controlling but actual temperature in the carb is, and Cessna did sell a carb temperature gauge as an option. It has a yellow arc in the temps where carb ice is most likely to form.
So, whether it's "ok" to operate a Continental powered 182 at max power all day, is debatable, but there's no limitation on it. You'll probably see cylinder head temps climb too high for most people's comfort unless your airflow and under cowl baffling is perfect and you may have to fly around with the cowl flaps open or partially so. Most folk won't do it for extended periods at low altitudes. Also fuel burn obviously goes up considerably since you'll fly a little faster and a 182 is nothing but a giant parasitic drag machine. So you'll burn a lot more gas to go about 7 knots faster, heat everything up more, and generally you're just flogging it.
Once you climb up (and you want to climb up, remember what happens to true airspeed as the air gets thinner) the engine won't be producing full power anyway, and you'll have to lean and you'll probably want the prop back into a "cruise" setting for slightly better efficiency at that speed. So you use the power chart and pick what you want. Prop even further back, slightly slower but even less fuel burn. It's like putting your car into overdrive on the highway.
Unless you're staying below 3000' MSL, you're not getting full rated power out of a non-turbo O470 anyway. If you do, watch temps carefully and expect 15 GPH burn. Climb up to 10,000 MSL where maybe at today's temp you're making 65% rated power and notice your true airspeed increase. And now you're going somewhere at about the same groundspeed for 11.5 GPH. Nice.
Not much point in flogging most normally aspirated engines. Climb up, enjoy the better view and more options for gliding to a safe landing if the engine quits, and all that.
There's a way to graph all of this out on a chart to show the most efficient way to go for short distances, long distances, etc... Because you lose time and speed in the climb and maybe (if no turbulence and you're willing to descend in the yellow arc on the airspeed indicator) get some of it back (not all, remember parasistic drag, that *****, goes up as ^2 not linearly as you speed up) in the descent. So on short hops, it's not advantageous to climb high. On longer trips it almost always is. Barring climbing into headwinds.
Eastbound I'll always climb up since I'm almost guaranteed a tailwind. Westbound, especially in winter, I may stay lower and flog it a little going upstream. But not usually 0-3000' MSL amounts of flogging. Just burns too much fuel for very little gain overall.
