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Let's assume a plane is initially moving at constant speed, level flight with a certain angle of attack. If the plane's thrust is increased, the plane's horizontal speed increases (at one point the drag force catches up and becomes again equal to thrust. The speed becomes constant at that point).

But while the speed is increasing, the lift would increase and would the plane start climbing. How can we keep the plane in horizontal level flight while the speed is increasing? Does increasing the thrust need to be simultaneously complemented with controlling some of the rear lift surfaces to oppose the natural lift increase due to the wing?

Simply put, in the scenario stated, to maintain level flight the appropriate amount of nose down input would be applied.

As the op points out, the increase in speed through the air increases lift, and the AOA (angle of attack) needs to be reduced. This is normally done with stick (or yoke) input, followed by trim.

On most aircraft the lowering of the nose would be accomplished by lowering the elevator (and/or raising the trim tab) on the rear horizontal surfaces.

Assuming a conventional, stable aircraft, most likely yes (unless the changed thrust moment around the centre of gravity balances the aircraft in trim in the new aerodynamic state).

„Normal“ aircraft are statically stable in pitch, meaning that a disturbance in either speed or angle of attack will lead to a change in aerodynamic forces towards restoring the old state. In your case, adding thrust will increase speed and hence lift, so the aircraft will naturally pitch up to restore its old speed and angle of attack (now in a climb due to added thrust). To prevent this and keep level, angle of attack must be reduced to keep lift equal weight at the higher speed.

Note the stability in angle of attack and speed is a simplistic view - real life might be more complicated, as the aircraft will most certainly enter some form of oscillation before (again, on an aircraft of conventional design) settling into a new state.

Some electronic flight control systems are designed to pronounce this self-restoring tendency to the trimmed speed (e.g. that’s Boeing’s Fly-by-wire philosophy), some other flight control systems are designed to suppress it (e.g. on a Fly-by-wire Airbus).

Hence (and slightly on top of what was actually asked), on Fly-by-wire Boeings and conventional aircraft, the throttles are more or less „up/down“ controls and the yokes are more or less „fast/slow“ controls, while on an Airbus it’s vice versa (and sticks not yokes).

Trim for straight and level just as you normally would. If you increase power you will need to retrim or apply nose down.

The increased airspeed, including the increased prop wash, also flows across the elevator (conventional aircraft), and for a typical (non-aerobatic) aircraft, the center of lift is a bit in front of the center of mass, and the elevator produces a bit of downforce to compensate, and the increased air speed over the elevator would also result in an aircraft pitching up.

For model aircraft, the prop axle is oriented a bit downwards (to reduce pitch reaction to increased thrust), and also a bit sideways (to reduce prop yaw effects). I don't know if full scale aircraft do this.

Compensating for more lift does not require more down force ... more lift while in stable flight WILL result in a climb, period. The compensation is to reduce angle of attack for the purpose of maintaining the same lift at a higher air speed. The horizontal stabilizer does not "push down" on the aircraft as a whole - it pushes up or down on the tail to control this angle of attack.

Engines do not control speed in an aircraft. Elevators control the speed. Consider a glider (sailplane) which can control its speed quite happily without any engine thrust.

Engines control the rate at which total aircraft energy, kinetic plus potential, is gained or lost.

Looking at the exact wording of your question, 'how to maintain level flight while accelerating' ...

Accelerating means increasing speed. To achieve this you push the stick forward. The increased speed means increased kinetic energy, which has to come initially from potential energy, so there is an immediate loss of height. The increased speed also means increased drag, so an increased rate of loss of energy and so height. To maintain height, that is, level flight, you would increase engine power to match the increased drag, and accelerate sufficiently slowly that the engine would also provide the increased kinetic energy.

Yes, an increase in airspeed creates more lift which requires more down force from the tail to keep the airplane from climbing. To maintain straight and level flight the total lift forces must equal total weight (1G). In order to compensate for the increased lift more down force must be generated to balance out the increased lift. So cruising at high speed puts more stress on the wings and horizontal stabilizer / elevator.