First off, be careful not to generalize too much with the races when it comes to weaponry - men were just as likely to use bows as elves, and the sword was actually the primary/most common weapon for elves. And while it’s true that dwarves tended to prefer axes, they weren’t the only ones to use them, and they were happy to use swords as well (as we see with Thorin and Orcrist.)
Middle Earth is (broadly) based on northern Europe during the “Dark Ages”, so right off the bat you can assume that any weapons common in that time and place were probably found in Middle Earth. The real question is whether or not they were common enough to be mentioned by Tolkien. (On that note, you see more variety of weapons in the Peter Jackson movies than are necessarily mentioned in the book. That makes these weapons not quite canonical, but I personally find it to be more realistic.) Here’s a list of weapons we see:
Axe: Axes were most commonly used by the dwarves (and were apparently used so frequently among the dwarves as to lead to the battle-cry “Baruk Khazâd! Khazâd ai-mênu!” - “Axes of the Dwarves! The Dwarves are upon you!”) But axes were also used by men. And the Sindarin elves of Doriath (in the First Age) favored axes for their own warriors.
Bow: Bows were pretty common in Middle Earth, since they were not only useful in battle, but also for hunting. And there are several characters known specifically for their skill with bows (such as Beleg Strongbow and Legolas.) We see different size bows, and even bows made of different material, throughout the stories. The elves of Lorien, and most men, seemed to favor longbows. Meanwhile, the elves of Mirkwood, and most orcs, seemed to favor smaller bows. Most bows are made of wood, but it’s said that the Numenoreans learned to make bows from tempered steel.
Crossbow: In Peter Jackson’s Two Towers adaptation, the Uruk-hai attacking Helm’s Deep were shown using crossbows. However, there’s no mention of crossbows in the books themselves, so I wouldn’t consider this to be canonical.
Dagger/Knife: Knives are usually secondary weapons, and as far as I can tell there weren’t any characters who used knives as their default weapon (with the exception of the hobbits, who tended to use “big folk” knives as if they were swords.) But for some, these offhand weapons were used very regularly (like Legolas, who often used his long knife when his bow was no longer useful.)
Spear/Lance: Spears were used mainly by men and elves. Gil-galad, the last High King of the Noldor, was famous for his skill with his spear Aeglos, which he fought with until his death in the Battle of the Last Alliance.
Sword: The sword is definitely the Great Weapon of Middle Earth. Every race used swords in some form, and for men and elves the sword was generally the most commonly used weapon. In fact, the elves were famed for their superior swordmaking skills (especially their ability to make magical swords that glowed blue when orcs were nearby), and the vast majority of the named weapons in Middle Earth are swords. Since shields are often mentioned alongside swords, it’s probable that most swords in Middle Earth were meant to be used with one hand. And, generally, the elves used straight swords, while orcs used curved blades (there were exceptions, though, like the fact that Egalmoth of Gondolin was said to use a curved blade, and the Uruk-hai were described as using short, straight swords.) And we also see swords made of several different materials - the earliest elves used iron, until they learned to make swords of steel. And one sword, Anglachel, was actually made from the iron ore of a meteorite. Tolkien actually doesn’t describe the swords too specifically, though (the most detailed he gets is in the elvish words he creates for “broad sword”, “short sword”, and “cutlass sword.”) So it’s probable that there was a much greater variation of types of swords that we just don’t see as readers.
Many individual weapons are incredibly important to the stories they appear in, and many characters are strongly associated with their specific weapon, so this is a really good topic for further research. If interested, check out this short essay on the history of weapons and armor in Middle Earth.
SOURCES: The Silmarillion, LOTR, LOTR Appendices, Joe Piela’s essay “Arms and Armor in JRR Tolkien’s Middle-earth”
((If you’ve noticed that I missed something, please let me know - this is a big topic, so I’ve no doubt that I missed something somewhere, lol.)
Egalmoth was one of the lords of Gondolin, the mighty hidden city of the First Age. We know he was Noldorin, but we don’t know if he was born in Valinor before the exile of the Noldor, or born in Middle Earth afterwards (to be a lord of Gondolin, I’d assume that he was born in Valinor, but we don’t know for sure.)
His physical characteristics aren’t described, but as one of the Noldor it’s likely that he had dark hair and dark eyes. He wore a blue mantle with crystal stars on it. He was very talented with a bow, but he also carried a sword - in fact he is the only Noldo known to carry a curved blade.
Egalmoth’s house was the House of the Heavenly Arch, which was a very wealthy house known for their many jewels. Their shields were blue with a jewel in the center built of seven gems (rubies, amethysts, sapphires, emeralds, chrysoprases, topazes, and amber), creating a rainbow effect. They mostly defended the northern wall of the city during its attack.
Egalmoth actually survived the fall of Gondolin, escaping with Tuor and a few survivors from his house. They made it to the Havens of Sirion, where they lived in relative peace until the Third Kinslaying (when Feanor’s sons attacked the Havens after Elwing refused to give them the silmaril.) Egalmoth was killed in this battle.
SOURCES: The Silmarillion, The Lost Tales Part 2 (“The Fall of Gondolin”)