Defining what makes a 'superior' weapon is somewhat difficult, as its effectiveness is dependent not only on functionality, but on logistics and employment. For example, the U.S. BAR was a deficient squad light machine gun, compared to the Bren or MG34/42, but this was not a failure of the U.S. arms industry, rather a doctrinal choice to focus on the rifleman (with a semiautomatic M1) as the focus of the squad, as opposed to seeing the riflemen as support for a machine gun. So you could say that the U.S. had a better infantry rifle and the Germans had a better machine gun, but that misses the point.

You mention the STG-44. This firearm is seen as special because it's more-or-less the first assault rifle, which type has become dominant for primary infantry weapons. However, this isn't really a technical achievement- the allied powers could certainly have fielded an assault rifle, had they identified a need for one. Rather, it was a doctrinal acceptance by the Germans that there needed to be a weapon in between submachine guns firing pistol ammo and rifles firing full-power rounds (i.e. an assault rifle). They had determined that most infantry actions did not occur at ranges where a full-power cartridge was needed, and the intermediate round in the STG allowed it to be used for automatic fire, and at ranges greater than the MP-40. So it was sort of a happy medium, but again, it was a doctrinal development, not a technical one.

Were there instances of U.S. troops using German weapons? Sure. Was it widespread practice? No. Again, it's about logistics and employment. If you give someone a weapon they aren't trained to use, that requires ammunition and spare parts that they have to capture as well, and moreover may have a very distinctive sound that would draw friendly fire, they aren't going to have a good time. So even if one side had had much better weapons, it wouldn't have behooved the others to use them.

So, what's the upshot for you? You could certainly give a lecture on all this weapons procurement jazz, or on squad-level infantry tactics, but I don't think that's what you should be doing. All these nitty details are important and interesting, but maybe not so much on a high school level. You could go for the typical Reddit wargasm of "Simo Häyhä could noscope Russians from a mile away and Audie Murphy had the highest killstreak of all time!", but, again, that isn't really valuable for understanding. If you link the anecdote into the history, i.e. explain that Häyhä got so good at shooting through prewar training with the White Guards, and that he was advantaged by the Soviets' early lack of winter camouflage, their poor training, and their officer corps that was gutted by Stalin, then you're on the right path to mix evocative narrative with deeper comprehension, at least to my mind.

as far as I'm aware they weren't the technological masterminds that movies and books paint them to be...were they?

Nazi Germany revolutionized tank and air combat. They were the first country to deploy radios in their tanks as standard equipment, and the first to include a tank commander position in all of their main battle tanks. This means there is a man in the tank who basically watches for the enemy, gives orders, and mans the radio. All other tanks separated these jobs onto people who were already doing other super important work like driving the tank, aiming and firing the main gun, or reloading/swapping ammunition. The russian T-34 is an example of a tank that went into a modern war using flags and hand signals. Germany also had extremely good glass and lense makers, meaning they had extremely good weapon sights for the tanks. This allowed german tanks to be effective at much longer ranges. They built an incredible fighter at the time in the ME/BF-109. It was fast, tough, had a radio, and was armed with a cannon in a time when most warplanes still used smaller machine guns. It was also armored. Through regular upgrades this plane stayed competitive through the end of the war. Other technology? Well, they did amazing things for rocketry as you mentioned. They did not (in my opinion) do nearly as well with small arms, which I will get into later.

In terms of the primary small arms of the varying forces, was there a definitive "superior" weapon?

You're going to get a different answer depending on who you ask but I agree with General Patton. The M1 Garand marked a paradigm shift in basic infantry capability and doctrine, and put a level of firepower at the most basic level of an army that no other army in the world at the time had. Every other country went to war with a bolt action, 5-6 round full size battle rifle (the british SMLE, the german KAR-98k, the russian Mosin-Nagant, the japanese type 99 Arisaka, etc).

The Garand put 8 rounds of rapid firing semi-automatic power in the hands of every rifleman. It could be reloaded extremely quickly with a "stripper clip" injecting a full magazine worth of rounds in a second. With the Garand, an 8 man rifle squad was able to generate the same amount of fire on a target as a machine gun grew, while being far more mobile and using the same ammunition. The US army was the only force on the battlefield capable of this during the war.

I would also say the russian Ppsh-41 is a standout. All armies developed a sub-machine gun (germany - mp40, america - m1a1 thompson, england - Sten, etc) but the ppsh stands above the rest because it does the 2 things you need an SMG to do better than any other option: high rate of fire and large (reliable) magazine capacity. The thing is literally a bullet hose that makes no pretense of trying to be accurate which is great for a weapon that was usually fired at point blank ranges, from the hip, while running.

Lastly it might be worth mentioning that the infamous AK-47 was a direct result of Russia capturing STG-44's and reverse-engineering/improving the design.

LMG or GPMP

The only thing we know with certainty is that the Mg 42 was the best light machine gun of the war, pretty much uncontestet. The M2 was/is a very good machine gun but doesn't really fit the same role as the Mg42 because of its size so its not a contender for ths role. The MG42 was cheap, rugged, mobile and accurate in relation to its rate of fire. It was easy to handle in every aspect. The German doctrine made the small units were heavy on the LMG side. Some weapons could give a bit more accurate fire but thats not what this weapons were about, they were about suppressing the enemy and causing maxium destruction and confusion over a specific area. The Mg42 was best at that and easy to manufacture. Some would argue the Mg42 isnt a LMG but a GPMG but...

Sub machine guns

The Red Army fielded powerful submachine guns even early in the war but other countries also introduced more submachine guns later and its hard to say which was "the best". Many designs had their merits. The PPsh 41 and the Mp40 should be mentioned here. The PPsh could be devastating in the support rule and close quarter combat because of its rate of fire and magazine the MP40 was pretty accurate easy to handle and to manufacture.

Semi automatic rifles

The M1 Garand should be mentioned as likely the best semi automatic rifle of the war and also as pioneer of said weapon class. But the edge here isn't as clear cut as with the LMG. The Gewehr 43 for instance has comparable characteristics.

HMG, even relevant to the question? Hm

For heavy machine guns the M2 is pretty alone in terms of quality, without a challenger really but it should be mentioned that other countries didn't focus on weapons in this weight region. The Germans for example were content with the Mg 34 and Mg 42 for supporting infantry and use in vehicles. This was in part due to different infantry tactics, the Wehrmacht prefered many LMG for their small units which was a highly successful concept. If they needed something bigger they used auto cannons 20mm and bigger. So in this rather small sub class the M2 is the best weapon. This weapon is still in use today which is pretty unique and speaks for the superb design.

Hand guns

There were a lot of hand guns and many of them featured some advantages over other designs i doubt you could establish one at the best...

Bolt action

The bolt action rifles of WW2 were often comparable at least for the countries fighting in Europe. If one design was better than the others it was marginally at best. Common contenders for the "best" bolt action are the Kar98 and the Enfield with more supporters for the Kar98 as far as i can tell.

Assault Rifle

The Sturmgewehr 44 was the first of its kind so...

Honorable mention the FG42 which was in itself a superb weapon but saw only limited use because of its development as niche weapon with a very high prize point. In terms of quality its hard to find a better weapon for a single riflemen than the Fg42.