Lancers
Other names for cavalrymen are be horse soldiers or mounted soldiers. I don't know of any in English beginning with C; you might use chargers.
Spahies / Spahees
The Battle of Little Bighorn saw 3,500 Cherokee cavalrymen fight for the Confederacy during the US Civil War.
Officers, cavalrymen, and artillerymen
The higher tier of the aristocracy were the patricians. The lower tier were the equites (cavalrymen)
No. Enlisted men had muskets and later breech-loading carbines.
Roughly 700 cavalrymen were part of the British Cavalry unit that made the suicide charge against the Russians in the poem Charge of the Light Brigade.
It is not clear what you mean by subcategories. In Roman society there were three social ranks, which they called orders: the patricians, the equites (equestrians, cavalrymen) and the plebeians.
From a wartime strength of about 100 men to a frontier post civil war strength of roughly 30 cavalrymen.
Lee intended to coordinate Pickett's charge with an attack on the Union rear by 8,000 Confederate cavalrymen led by General Jeb Stuart. There was only one problem--twenty-three-year-old George Armstrong Custer, a general for three days, denied Stuart the pathway to the Union rear by leading several thousand Union cavalrymen on a series of brazen charges.
All Federal cavalrymen were Union soldiers, but not all Union soldiers were cavalrymen. Most Union soldiers were infantrymen - foot soldiers. There were also artillerymen, who worked the cannon. Cavalrymen were horse soldiers. They rode horses where they went, and practiced as a group to train the horses to do the maneuvers they had to perform. They scouted ahead of and on the flanks of the army, they watched for the enemy to approach so their army would not be caught by surprise, and they sometimes tried to charge in a group to break the enemy line. The cavalry charge had been very important in Napoleon's time, and all American officers had studied Napoleon, but in the Civil War the foot soldiers had rifles (they had only short range muskets in Napoleon's day) and could shoot the cavalrymen at long range when they started a charge, so this was not done much. Cavalrymen usually had a type of sword called a saber, to cut down the enemy with in a charge. By the end of the war they had quit carrying sabers and started carrying six-shot revolver pistols, usually two of these per man. The cavalry's primary weapon was a carbine, a shorter rifle than the infantry used, which could not take a bayonet. The bayonet charge on foot still decided many battles, so if cavalry got into a dismounted fight with infantry, since they did not have bayonets, cavalry usually could not stand up long to infantry - they would have to mount their horses and ride off.