I’m a noob so sorry for my ignorance. The question is: do we need (in jet airplane) to increase the throttle to maintain constant thrust output if we want to fly higher (not considering climb phase), I mean whether on 2000 ft and 20000 ft altitude throttle is in the same position or we need to push it forward if we fly at higher altitudes to maintain constant power (as in piston airplanes).
It depends, as so often.
With the thrust available at low altitude, the indicated flight speed will be much higher than at high altitude. There, the reduced density and thrust will only allow flight close to maximum thrust and over a small speed range around the minimum drag polar point. In practical terms, at low altitude the airliner will not be able to fly level at the maximum thrust setting because it will exceed its Mach limit. But since the speed of sound is higher in warmer air, it will still fly significantly faster than at high altitude when it flies at the same Mach number at both altitudes.
If, on the other hand, you compare states at the same indicated air speed, the required thrust at low altitude will be a bit smaller due to the higher Reynolds number in warmer air. On the other hand, fuel flow will be a bit higher since the engine is not running at its optimum design point and the higher ambient temperature reduces its thermal efficiency.
While thrust will go down with air density (or dynamic pressure, for the same Mach number at low and high altitude) in the first case, in the second case it will increase slightly.
No. Power - Thrust or torque - on an air breathing combustion engine varies almost exactly in proportion with air density, so you will need every bit of power in a climb - full power all the way, engine temp limits permitting. A good rule of thumb is that every 15,000' the density halves and so does your power (unless it is a piston engine with forced induction, but power will still drop off). By 51,000', the highest certifiable altitude for (subsonic) civil aircraft, the density is 11%. This means your engine produces 11% of its sea level thrust. Granted the air resistance is 11% as well, so you need less power. Jets will burn all their fuel in minutes at low altitude, but can cruise for long periods at altitude, preferably at 40,000' and higher. The most powerful aircraft can take off with a very aggressive climb rate, even vertical for fighters, but this will drop off to a shallow crawl at higher altitudes when density and power drops. Forced induction is the answer for piston engines. For jets and turbo props, they are designed with high compression ratios.
As for wondering about changing the throttle setting - it is common for turboprop engines to get "flat rated", which is like a governor. It is automatic, but in a sense, this could be thought of as gradually increasing the throttle setting. Flat rating an engine limits their power at lower altitudes where it is not needed as much, and it also keeps the engine operating at lower limits throughout its life, which allows for longer overhaul intervals and cheaper overhauls. For example, you could have a 1200 hp engine that gets flat rated to 800 hp and it will simply operate at 800 hp in takeoff and climb till, say, 10,000' and then the power output will start to drop off after that.