They really did indeed use child soldiers.

Here you can see Hitler himself handing out medals to a group of child soldiers for destroying tanks in the battle of Berlin: https://www.awesomestories.com/images/user/4893b71ca0.jpg

This is considered to be the very last time Hitler was ever photographed outside his bunker.

Child soldiers were not limited to the battle of Berlin, either. Here is a photo of Willi Hubner receiving an Iron Cross for his actions in battle in East Prussia.

https://s-media-cache-ak0.pinimg.com/564x/b0/0a/c0/b00ac052e416a1c7bf83165fffec3ee6.jpg

This was not really a "Grassoots" effort - it was practically mandatory for German boys to be in the Hitler Youth organization during the war. In the Hitler Youth they would receive military training and often be used as flak helpers and such even before the war turned bad for the Germans.

Once the borders of the Reich were crossed by the Soviets then Hitler Youth units were often mobilized to fight against the invaders. You'd also see young boys in the Volksturm - this was a militia made up largely of old men and young boys or other males unfit for military service. Hitler Youth and Volksturm units did not really serve in the conquered territories but were almost exclusively used to defend Germany itself as the Allies advanced into it.

There are exceptions of course - the 12th SS Panzer Division fought in Normandy against the Allies, and was known as the Hitler Youth division. The officers and NCOs were veteran SS men, but the average rank and file was made up of teenagers, though not so young as the 13-15 year olds you'd see defending Berlin.

Using child soldiers was definitely a desperation move but it was definitely sanctioned from high command - as you can see from the photos, Hitler himself clearly approved of it.

While child soldiers are obviously not as effective on campaign, they sometimes do have the 'advantage' of being indoctrinated more fully than their older peers and more likely to follow suicidal orders.

The Nazis even planned for there to be an underground army to resist the allies in post-war Germany known as the Werewolves, which were largely made up of teenagers. In actuality the Werewolves never did much besides a few isolated incidents, and the organization was never as comprehensive as the Nazis claimed or hoped for in their propaganda efforts.

Germany employed many underage boys during the war. The scale of child soldiers increased during the war due to changing war realities. At first, they were used in more support roles and eventually they were used as full soldiers or "cannon fodder" as you pointed out. This started with "noncombat" roles like Flakhelfer or Luftwaffenhelfer where German boys age 15 or higher were employed to support anti-aircraft batteries and sometimes operate the Flak guns. Later in the war, children were often operating anti aircraft batteries without supervision. Not really noncombat but on another level than end war duties. The boys were used to defend their home cities. Some girls severed as well. Interesting here is that the young Flakhelfers were considered highly effective often more so than older soldiers. Given the sheer size of the programm you can expected several hundred if not thousands were killed during the war. Random hits by Allied bombers occured so did accidents.

This was the "first phase" of child soldiers in the Wehrmacht. The intermediate phase would be the 12th SS Hitleryouth, a unit which employed boys/men who excelled at their Hitleryouth service. Beginning in 1943 the unit was created using the best from the Hitleryouth pool. The officers and most of the NCOs were veterans of other units, a big part of the infantry and tank crews was 16-17 when they started training and 17-18 when they engaged Anglo/Canadian forces in Normandy. While morally abhorrent the concept payed off and the unit performed outstanding, showing the benefits of indoctrinating children in young age and preparing them to be soldiers. The 12th SS was maybe the least questionable action given that they were mostly 17 when thrown into combat, this would change later with soldiers getting younger and younger. 15 year old soldiers were common during 1945.

I think this picture illustrates the dire future of boys in Germany at the time. Young boys are presented soldiers from the 12th SS, their "future".

From this point on more and more underage boys were drafted to bolster the manpower of the Wehrmacht. Most of them in the "Volkssturm" but also many of them to regular Heer units. This can be seen for example during the Battle of the Bulge with very young looking units. 15 years were rare in Heer units used for offensive operations obviously.

Extremely young soldiers like during the Battle of Berlin only started to appear in the last months of the war when the war was brought to the Germans. During 43/44 you wouldn't see such young soldiers. When the Red Army forces started to advance on German soil boys from the surrounding areas would get "drafted" into service and then thrown against the attackers together with old people.

While German "Kindersoldaten" were often at least somewhat trained due to their experience in the German youth system and short courses, they were more or less cannon fodder. The exception here is the 12th SS because they unit had longer training time and an elite cadre.

As far as i am aware fighting girls were rare but there certainly were some. Younger children especially girls were often brought away from those areas long before the Red Army arrived which makes even spontanous drafting unlikely.

Numbers to casualties of those child soldiers are not known.

Was this, lets say, part of the Nazi German military doctrine or was it merely a last resort

This is a complex question but to give you a short answer, the entire German system was geared to be "combat ready". Young boys were trained in marching, discipline and shooting from young age and girls were trained to be good "breeding stock". Obviously it wasn't part of the "doctrine" since drafting such young soldiers wasn't planned. But the Hitler youth in a sense was already a military unit trying to bond German boys together teach them comradery and prepare them for the possibility of becoming soldiers. While the necessity to use child soldiers was not really expected, the youth system was designed to create the foundation for perfect future soldiers. These effects were shown in the early war where many of the "victorious" German soldiers were part of the Hitler youth and this culminated in the 12th SS, a unit with far less training time than an average Allied unit, heavily outperforming seasoned enemy units during ground combat. The idea wasn't to use child soldiers but to prepare children to become soldiers.