Hypnos,
Some principles you will read about regarding high performance airplanes and turbojets will apply to the flying you're doing with your warrior, but some won't. You ask some great questions, and you're to be congratulations on expanding your understanding...that's a excellent habit as a pilot.
Climb thrust has several connotations, especially in turbine equipment; it has to be put in context. During takeoff, the thrust that's set depends on a number of factors. Thrust can be limited by several factors, depending on the type of engine and the operating conditions, too. It's a little more complicated than a piston engine where takeoff is nearly always done at maximum RPM and throttle setting, on a fixed-pitch, normally aspirated (nonturbocharged) engine...such as the warrior.
In a turbine engine, reducing thrust for takeoff is a common practice. This is done for a number of reasons ranging from noise abatement to increasing engine life...so long as the takeoff can be made safely and all the necessary climb performance can be met. Takeoff profiles typically involve a couple of different profiles, but are similiar in that at some point after takeoff, climb thrust is set. Takeoff may be made with maximum thrust, or at a reduced thrust setting, and setting climb thrust after takeoff may mean a thrust increase, or decrease in thrust.
A typical takeoff involves a climb to 1,000' where climb thrust is set, using a speed about ten knots above the takeoff safety speed (known as V2). The climb is then continued to 3,000' above the departure field elevation at that speed and that power setting. Keeping that power setting, the nose is lowered slightly, vertical speed reduced to five hundred or a thousand feet per minute, and the airplane allowed to accelerate and the flaps to be retracted.
During the climb, several types of climb thrust may be used. Maximum climb thrust will be calculated for the climb, and often a reduced number somewhat below that will be used. Just like on takeoff, an engine may be limited by the maximum allowable temperatures internally, or it may be limited by other power settings such as torque, EPR (engine pressure ratio), etc. The maximum value that limits the engine may be used for the climb, or a lower number may be calculated for the climb. This is sometimes called climb thrust (the power setting to be used for the climb, as opposed to the maximum value that could be used for the climb...which would be maximum climb thrust.
To further complicate the matter, several values for the climb thrust may be available, limited by time. The maximum thrust is often called maximum continuous thrust, whereas other values may be allotted to the engine for shorter durations, such as five, twenty, or thirty minutes. Additionally, emergency thrust is available for situations such as a windshear encounter, and go-around thrust...all having different values, all being maximum thrust situations for their intended use...and all of them being climb thrust values. A go-around or missed approach, for example, will often involve a reduction of thrust as the airplane climbs out, from go around thrust to climb thrust...even though both are valid climb values.
Reducing thrust just a little has a very large influence on the longevity of a powerplant...just a few degrees or a few percent reduction can increase engine longevity by ten percent or more...or in economic terms, millions of dollars. To say nothing of increasing the safety margins by which the engines operate by decreasing thermal stresses and damage to the internal components.
Turbine engines are most efficient at high altitude because they must spin faster to produce the same thrust. As a turbojet climbs, the engine speed increases, and with an increase in engine speed, up to a point, engine efficiency increases. At lower altitudes, the engine produces too much thrust with the power pushed up that far...at 6,000' for example the airplane is limited to 250 knots, and pushing the power up too far will easily push the airplane faster than that. Accordingly, the engine must be operated at a much lower power setting to keep the speed in check (and to respect other airspeed limitations such as gear, flap, or even Vmo limitations)...the engine isn't being operated nearly as efficiently.
At higher altitudes, true airspeed increases while indicated airspeed decreases. The airplane can fly faster and faster with less and less drag, but the engine must turn faster to do the same job...to compress the thinner air and to produce the same thrust...the engine operates more efficiently.
The way power is measured or described in the cockpit really varies...in the airplane I'm flying right now, our chief power gauge is the EPR, or engine pressure ratio gauge. It tells us about the difference between the pressure at the front of the engine, and the pressure at the back...an EPR ratio of 1.0 means that the pressure at the back of the engine is the same as at the front...push the power up above that and the EPR starts to climb. An EPR of 1.5 means that the engine is pressure at the back of the engine is greater than the pressure at the front...it's producing thrust. The actual value isn't important, though it's in the neighborhood of about 60,000 lbs of thrust.
As we climb, we can calculate, or the airplane will calculate, the climb thrust setting at any given altitude...the EPR setting required increases as we climb to higher and higher altitudes. We may have to move the thrust levers to meet this value, or the value may increase all by itself...for a given thrust lever position/power setting, the EPR increases as we climb. All we need to do is tweak it...and that might mean pulling the power back slightly to keep within limitations, or pushing it up. Very often we will do a reduced thrust climb. We do this once we reach 10,000 during the climbout, and for us it's a simple process. We reduce our EPR by .04 if we're heavier than 600,000 lbs, and we reduce it by .06 if we're 600,000 lbs or less. We will maintain this reduced thrust until our climb rate drops to 500 fpm; this typically happens about 27,000' to 30,000'. At that point we increase thrust to the maximum (I usually keep it .01 or .02 low to keep from exceeding any limits), until we reach our cruising altitude. From then on, we keep the thrust where it's needed in order to maintain .84 mach. (This, incidentally is generally well below the maximum cruise settings).
If one goes too high, one is operating less efficiently. We find that cruise at the optimum altitude isn't always possible. Generally cruising just a little below that is preferable to trying to climb above it...we end up having to carry too much power and fly at too high an angle of attack (with too much drag) if we're above our optimum altitude. I mention this because there's a lot more to choosing a cruise altitude when operating efficiently than simply taking the engine as high as it can go.