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We all know the 2 stages LM design used by Grumman was intended to discard the mass of the landing gear (+ other components) at the moment of launching off the Moon surface to reach back the Service module. But was it really necessarily for the LM to have two stages? The reason I wonder is that, when Armstrong landed, as we know there was about 25 seconds fuel left - however, this was actually 25 sec. of fuel before aborting the landing with the complete LM, not before running out of fuel. After these 25 second would have ended, the LM still had enough fuel to ascend with both of its stages right back to the Service module. In other words, the LM was designed to be able to take off from the Moon surface with BOTH stages, even right after touching the surface, in case something would have gone wrong. Then, why using two stages which surely added complexity, weight and a second engine?

After these 25 second would have ended, the LM still had enough fuel to ascend with both of its stages right back to the Service module. In other words, the LM was designed to be able to take off from the Moon surface with BOTH stages, even right after touching the surface, in case something would have gone wrong. Then, why using two stages which surely added complexity, weight and a second engine?

Your assumption is not correct. Aborting from the "bingo" (low fuel) call would have required the ascent stage to be used. The stages can be separated, and the ascent engine fired, while in flight; this was demonstrated on Apollo 9 and Apollo 10.

Because there would be a brief delay between staging and the ascent stage coming up to full thrust, the safest way to abort in this case would be to take the descent stage to full thrust to gain altitude and vertical speed, then stage and activate the ascent stage engine once the descent stage fuel was exhausted.

The ascent from lunar surface to rendezvous orbit took about 7 minutes on the ascent stage; there was nowhere near enough fuel in the descent stage to do that.

Early conceptual designs suggested that two stages would save weight. Another issue came up that made a single-stage lander unreliable.

Pressurizing the fuel and oxidizer tanks of the lunar module was a considerable engineering challenge. Because of the temperature changes that happen during launch and Earth orbit, the tanks were kept unpressurized until right before they were needed. At that time, the astronauts would fire some explosive valves (i.e. you can open them but never close them again) which would release a small amount of supercritical helium to pressurize the tanks.

Development and testing of the LM pressurization system took 6 years and had a lot of problems. The fuel froze, so they added a heat exchanger. A test article exploded, so they had to change the alloy used in the heat exchanger. The pressure regulator cracked. You were supposed to be able to raise or lower the thrust of the descent engine; the pressurization system wasn't compatible with that, so there was another re-design. Another heat exchanger was added. The descent engine shut down prematurely on Apollo 5. It was under-pressured on Apollo 9. When Apollo 11 dumped their excess descent fuel, the fuel lines froze and over-pressurized.

Suffice to say, once you've popped the valves on the LM pressurization system, you have a few hours to use the engine before it becomes unreliable. This was enough time to land on the moon, but not reliable enough to last several days on the moon. A one-stage craft is just too risky. Instead, the ascent stage had its own fuel and pressurization system. (Apollo 13 took the risk of reusing the descent engine, and was lucky.)

The Soviets designed a manned lunar lander, the LK. To save weight, it would have had a single engine for descent and ascent. However, it still would have left some parts behind on the moon.