Cataphracts seem to have originated in Central Asia, and were notably used extensively by the Parthians and the Sarmatians. According to Phillip Sidnell's "Warhorse: Cavalry in Ancient Warfare" the earliest attested unit of catafractarii in the Roman Empire was the ala I gallorum et pannoniorum cataphractaria. The Strategikon of pseudo-Maurice gives heavy insinuation that cataphract warfare was adopted from (and as a counter to) the Sarmatians and Alans on the Danube. The Romans had perfected cataphract warfare in the late 6th century and would continue using "Cataphracts" until the early 14th century.

To your other points:

1. Roman units of Kataphraktoi were paired with horse archers, and were supposed to be armed with bows themselves according to the 9th-11th century military manuals (the Peri Strategias, Taktike of Leo VI, Sylloge Tacticorum, Praecepta Militaria, and the Taktike of Nikephoros Ouranos). They quite thoroughly detail how to arm the horse archers integrated among the kataphraktoi, who were usually formed in a square block or a wedge.
2. Evidence from the notitia dignitatum (398-425ish AD) suggests that many professional regiments of catafractarii and clibanarii originated as symmachi ("allies"). For example the comites alani, the equites prima clibanarii parthi, equites persae clibanarii, equites secunda clibanarii parthi, comites cataphractarii bucellarii iuniores, and equites quarta clibanarii parthi all suggest foreign recruitment or even origin.
3. The best books to introduce you to this topic are probably Warhorse: Cavalry in Ancient Warfare by Philip Sidnell, Roman Heavy Cavalry I: Cataphractarii and Clibanarii by Dr. Andrei Negin and Dr. Raffaele D'Amato (take Osprey books with a grain of salt, but they actually do a good job of going over the scholarship in this book). The Strategikon and the later middle Byzantine period military manuals are also all useful for learning about Roman cavalry warfare, which they detail extensively. Their translations can be difficult to afford however.
4. In cultures where arms were supplied by your own person, yes they would have to be in order to afford that kind of equipment. In the Roman army? No, the Romans were a professional army and their equipment was supplied by the state or they received an allowance to purchase equipment from the state. A peasant could be a catafractarius in the Roman army if they showed the aptitude for it.
5. Most likely. Evidence for women warriors after the Scythian period (8th-5th centuries BC) declines significantly, but we have finds of possible early Parthian women who fought as heavy cavalry that have recently been excavated. There is at least one example of an Alan-era lancer who was female from the Crimea as well, and the archaeological evidence disappears after the 3rd century AD. But textual evidence remains (although it can sometimes be unreliable). Of course, this probably does not apply to more "Traditional" settled cultures (the Sassanids, Chinese, Romans, etc.)
6. I mean... yes they could be all three depending on the situation. Shock lancing tactics were what they specialized in but they could also serve as very heavily armored archers and also fight with smaller weapons as well.
7. I don't think there's enough scholarship on this topic to effectively answer this. Large military horses like we think of today become a thing in the high middle ages (late Viking and Crusades period). The Romans, and other classical and medieval peoples, rode what amounted to "ponies." Archaeological evidence suggests Przewalski's Horse, Tarpans, and other varieties of steppe pony were common among peoples like the Alans and Sarmatians.
8. The Roman military authors don't comment on this really, and they seem to imply the opposite; the cavalry should take the priority in receiving armor, which makes some basic logical sense since the infantryman's primary defense was his shield, which cavalry did not rely on to the same extent.
9. There's too many factors to give an honest answer to this. Assuming all other factors are equal the biggest one always ends up being the skill of the leadership.
10. Knightly chivalric codes didn't apply to the Roman military or other cultures (and were often ignored by the western Europeans themselves). Cataphracts who fell in battle would have typically been dispatched afterwards and then looted for their armor and weapons, rather than ransomed.
11. We don't see a lot of instances of dismounted cataphracts, Tacitus briefly discusses how the Sarmatian cataphracts were hampered and slipped and fell under the weight of their own armor fighting in icy conditions in the 2nd century AD. See this paper here: https://www.academia.edu/28156928/Sarmatian_Armour_According_to_Narrative_and_Archaeological_Data
12. Generally speaking your average cavalryman rode 2 to 3 geldings (neutered horses). The Romans of the middle ages say that the kataphraktoi would have pretty much every weapon available to them: either a long sword (spathion) or saber (paramerion), an axe/warhammer/mace (or even all three), and a bow and quiver, as well as a lance of course. They would have had a single set of armor (there was no need for multiple), at least within the context of the Roman army of late antiquity and the middle ages.

I hope this answers your questions. Please let me know if there's anything you would like more detail on, I can start pulling additional articles and books for more information.