I'm not a historian, but I am an engineer working in advanced manufacturing of metal components with a graduate degree (and professional experience) around metallurgy.

I can't answer about sandpaper here, but I can tell you a little bit about files.

First, why does a file even work? Why does a file cut a part and not the other way around? It's not just about the rough surface— try rubbing a copper file on a steel bar and see how far you get. Files (and material removal processes generally) depend on the relative hardness of the two materials, with the harder material removing bits of the softer material. This is highly simplified but we'll leave the technical aside there since this is /r/askhistorians and not /r/askscience.

The question you're really asking is about the hardening of steel and its application in tool making. The earliest written description of these processes was compiled around 1100-1120 by Theophilus in his treatise "On Divers Arts" [1], where he writes fairly detailed instructions on a variety of artistic and industrial techniques (painting, glassblowing, metalworking, etc.). In it he describes case hardening a part via carburization, which is the process of adding extra carbon to the surface of a low-carbon steel, allowing the surface to become hard and brittle while the core remains soft and ductile. [Edit: see /u/xeimevta's response below for excellent context and quotes from Theophilus]

The basic process is to cover a low-carbon steel (read: wrought iron) part in carbon-containing material (charcoal, burned bone, old fat), heat it in a low-oxygen environment (achieved by sealing in a clay envelope), and then quench the hot part in liquid to rapidly cool the surface. Carbon atoms diffuse into the surface of the part, making a thin layer of high-carbon steel, which is then hardened during the rapid quench. Smiths and artisans of the middle ages understood how to carburize but did not understand the process mechanistically, making it a bit of a magical art. Myths abounded in this environment, as described by Theophilus, who is adamant that the best quenching liquid is "the urine of a red-haired boy".

So to directly answer your question about file making in any period up to the Renaissance:

1. Roughly form the file shape from raw wrought iron, using a combination of forging and grinding on stone wheels.

2. "Strip" the surfaces of the file to be completely flat

3. Form the teeth using a chisel on each cutting surface of the file (the chisels are made from the same steel and hardened prior to this process)

4. Harden the fully formed files via carburization, quench in liquid, and then (optionally) temper to the desired final hardness.

I highly recommend checking out Clickspring's Youtube channel, but specifically the two videos I'll link below. For the past 4-5 years he's been working on the enormous project of reverse engineering the Antikythera Mechanism from detailed CT scans of its internal mechanisms, and then manufacturing one using as many period-accurate techniques as possible. As part of this project he has performed original research on tools and techniques, and has co-authored a paper currently under peer review. These should illustrate the answer to your question on files.

Antikythera Fragment #3 - Ancient Tool Technology - Hand Cut Precision Files

Antikythera Fragment #4 - Ancient Tool Technology - The First Hardened Steel

I know that /r/AskHistorians isn't big on Youtube sources, but in this case I feel it should be allowed, it's shockingly on topic. /u/ClickSpring, it doesn't seem like you're still active, but please feel free to correct me on anything here.

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[1] Hawthorne, J. G. and C. S. Smith Theophilus: On Divers Arts. University of Chicago Press, 1963; ISBN 0-486-23784-2

I'll try to answer from a metalworking perspective. My expertise is in precious metalsmithing, but similar methods would have been used for processing copper, iron, and so on.

I can't really speak to the invention of sandpaper. I've only found anecdotal evidence so far. I can tell you, however, that metalworkers in medieval Europe and Byzantium did not use it. One of the best primary sources for medieval metalworking is the twelfth-century Schedula diversarum artium, or "On Diverse Arts" written by a Benedictine monk under the name Theophilus Presbyter. There's been a lot of recent scholarship on the authorship and motivations behind the composition of this text, but the scholarly consensus is the author was a working artisan, so he is considered very reliable. The most famous translation of the work into German was conducted in concert with Erhard Brehpol, probably the father of modern goldsmithing. Brehpol also did a lot of reconstruction of Theophilus' directives.

The third book of the Schedula is devoted to metalworking. He discusses files in the opening of the book, in a larger discussion of tools and workspaces:

Files Hollowed Underneath

Iron tools are also made as thin as a stalk, one finger in length, and with four sides. They are broader on one side than the other, and the tangs, which are fixed into handles, curve upwards. Underneath, a groove is chiselled out throughout their length and filed like a furrow, and the edges on each side are filed sharp. With these tools, coarse and fine wires of gold and silver are filed so that beading appears on them.

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Rasps

Iron rasps are also made, which are thin but rather wider at the end and sharp on one side. Some are small and some large, and some of them are curved back as required by the nature of the work. Tools are also made, shaped in the same way, but blunt, for burnishing the work.

I'm personally still trying to work out what exactly Theophilus is describing as a hollow file, but the iron rasps equate to modern-day hand files. In this same section on rasps, Theophilus notes that they also use burnishers for fine finishing.

Medieval goldsmiths made these tools themselves, and Theophilus tells us how:

Files

Files are also made of pure steel—large, medium, four-sided, three-sided and round. Others are made so that they are stronger in the middle, of soft iron inside but outside covered with steel. When these have been beaten to the required size, they are levelled off with a straightedge and then are incised with a hammer which is sharp on each side. Others are also incised with the chisel which we have spoken about above. These are used to smooth out the work, which has first been filed with the other coarser files. When they have been incised on every side, proceed with their tempering in this way.

Tempering the Files

Burn an oxhorn on a fire and scrape it; mix with the scrapings a third part salt and grind vigorously. Then put the file in the fire and, when it is white hot, sprinkle the preparation over it on every side. Open out the white hot coals and quickly blow all over, but seeing that the tempering does not fall off. Immediately withdraw the file and quench it evenly in water. Take it from there and dry it gently over a fire. In this way you temper all files made of steel. Similarly you make small files, square, round, half- round, triangular and flat ones from soft iron and you temper them in this way. When they have been incised with the hammer, or chisel, or with a knife, smear them with old hog’s lard, bind them round with straps cut from goat-skin, and tie them up with flaxen thread. Afterwards cover each one separately with kneaded clay leaving the handles bare. When they are dry, put them in a fire and blow vigorously until the skin is burnt. Then remove them quickly from the clay, quench them evenly in the water, withdraw them and dry them at the fire.

Over the course of this book, Theophilus recommends a number of different abrasive tools and substances depending on the desired result and material. These include soft ground stone mixed with spit and applied using wood or leather, sandstone applied with water, whetstones, crushed potsherds, and crushed tiles and charcoal. One of the great things about Theophilus is that, although he's primarily concerned with precious metal objects for use in the liturgy, he also remarks on ironworking, tinworking, and founding. So we can extrapolate from his knowledge of tools and materials out to other kinds of smithery.

Theophilus is a cornerstone of this area of study in the medieval period, but there's some exciting work happening among non-western medievalists right now pertaining to similar tool construction and techniques in Byzantium, the Islamic world, and China. A lot of similar techniques were used outside of Western Europe, and it seems like exchange and circulation of metalworking knowledge was extremely common across cultures in the medieval world.

Some further sources:

Dodwell, C. R. The Various Arts. De Diversis Artibus. Oxford: Clarendon Press, 1961.

Gearhart, Heidi C. Theophilus and the Theory and Practice of Medieval Art. College Park: Pennsylvania State University Press, 2017.

The Greek Treatise 'On the Highly Appreciated and Famous Art of the Goldsmith' - Edition and Interdisciplinary Commentary, a project out of Mainz.

Also a source: me, a medievalist and practicing metalsmith!