Before refrigeration and modern transport, the seasonal availability of many foods and restricted regions of origin made preserving food important. Before canning, the key methods were drying, salting, fermenting, and smoking. These were often used in combination (and smoking almost always in combination with one or more of the other methods). Drying was limited by climate and weather, and salting by the availability of salt (which depended on trade networks, availability of deposits of salt that could be mined, and seawater or other salt water and climate/weather). Fermenting could be done anywhere, and was widespread. Where drying and salting were feasible, it was often combined with them, but it was also used alone.

Many traditional fermented products are still in modern use, such a yogurt (fermentation alone, and with the benefit for the lactose intolerant of reducing lactose content), sauerkraut and kimchi (fermentation + salting), salami (fermentation + salting + drying to varying extents), and fish sauce (fermentation + salting), to name just a few popular examples.

There are also many regional specialties, more or less known outside the places where they are produced and eaten. For example, in the Faroe Islands, seawater aplenty was available, but salt could not be produced from it by evaporation due to the climate (which also made drying slow or impossible) and pre-modern trade was infrequent. Firewood was not available for smoking. How then to preserve mutton? The answer: fermentation, by hanging a leg of mutton in a ventilated shed. After 6-9 months, it is preserved and edible (and, reportedly, tasty but an acquired taste). The making of this, skerpikjøt, and its eating, is described by Nilsson (2017). Nilsson also describes the making of kiviak in Greenland, where a sealskin is stuffed full of little auks, sealed, and left to ferment for a few months.

Closely aligned with the topic of fish sauce, fermented fish products are quite common. The type of fish sauce still common in SE Asia was once much more widespread, and was a common sauce in much Chinese cuisine 2000 years ago. Later, when soy sauce was developed, it displaced fish sauce in East Asia, but not completely, with fish sauce still being made and used in Japan, and it is still widely used in Korean cuisine, even if not as much as soy sauce or bean paste. The oldest historical mention of East Asian fish sauce I know of is the Neizi chapter of Liji, the Book of Rites - the oldest surviving versions date from about 200BC, but the original text is older. Further west, we have from the same time, a Mediterranean tradition of fish sauce, with garon is use in Greece already in the 5th century BC. For example, Athenaeus notes it in his Deipnosophistae:

Garon: Cratinus has this: "Your pannier will be chock full of garon." Pherecrates: "He has fouled his beard in the garon."

This ancient Mediterranean tradition has not completely disappeared, with the Italian anchovy sauce, colatura di alici, still being made (I haven't tried it, but it looks like SE Asian fish sauce).

A wide range of traditional fermented fish products are still made in India and nearby areas (Majumdar, 2016; Thapa, 2016). These illustrate the broad range of fermented fish products, which include fermented whole fish and fish steaks that mostly remain intact and look much like the original product, solid cakes of fermented fish (e.g., Cambodian prahok and SE Asia belachan), pastes, opaque liquid sauces, and clear liquid sauces. The relationship between the solid cakes and the clear liquid sauces is often like that between solid soybean pastes and soy sauces like traditional Korean soy sauce and Japanese tamari, where the liquid drips from the cake during fermentation. Other liquid sauces are made by fermenting in a liquid state (similar to most modern soy sauces); clear versions can be produced by filtering. Both solid/paste products, and opaque and clear liquid sauces were used in the Mediterranean. Grainger (2011) suggests that the Roman garum and liquamen were, respectively, opaque and clear (while noting that others don't agree). Roman allec was a solid/paste, made from the solid residue of garum and liquamen making.

We can also note that some fermentation is not for preservation, but serves other purposes. For example, dough can be fermented for flavour, as with West African kenkey maize balls. More commonly, dough is fermented to produce carbon dioxide to make fluffy bread. Fermentation is used to produce alcohol (e.g., beer and wine) and vinegar. Sometimes, fermentation is used to make an otherwise inedible item edible, such as Icelandic fermented shark, hákarl, where the fermentation reduced the excessively high urea content of the flesh.

1. Is there any evidence of trade goods traveling (one way or the other) between the Roman Mediterranean and SE Asia?

We know some details about Roman-Indian trade, and since India was trading with SE Asia at the time, some SE Asian products may well have gone on to Rome. We know of now specific examples. We do know that Romans visited SE Asia, in the 2nd and 3rd centuries AD, but this cannot have been responsible for spreading fish sauce from one of these regions to the other, since it had already been in use for many centuries.

2. Do we know if fish sauce was ever a thing in the intermediate areas of Persia or India?

Apart from the many Indian, Nepali, etc. examples described by Majumdar et al. (2016) and Thapa (2016), there is also Iranian mahyawa, a spiced anchovy sauce which is also widely used in the Gulf countries.

There may well be a connection, but it's lost in the prehistoric past. Perhaps diligent archaeology, such as has been applied to trying to track the origin and spread of wine and beer by analysing residues in containers, might yield some results, but wine appears to be more archaeologically fashionable than fish sauce, which is essentially ignored other than the attention paid to Roman garum.

We shouldn't think of Roman garum and SE Asian fish sauces as bizarre local foods, so unusual that they can;t have developed independently. The fermentation of fish is something that happens when the fish is left to itelf - the fermentation itself doesn't need to be invented. How to use that fermentation to make an edible and tasty product is something that needs to be invented, but given the many different methods that are used for fermenting fish and meat products around the world, there are many different solutions that have been found. Rather than strange foods, these sauces are representatives of a broad range of traditional fermented fish products, and an even wider variety of traditional fermented foods.

3. How similar is SE Asian fish sauce to garum, as far as is known? Is it a decent stand in for ancient garum?

Roman garum appears to have had about half the salt content of typical modern SE Asian fish sauce (Grainger, 2011). Modern SE Asian fish sauces appear to usually have about 8,000-10,000mg of sodium per 100g; Italian colatura di alici about 6500mg per 100g (based on 1 brand only).

Apart from the higher salt content, it's a fine substitute. Colatura di alici, which you might be able to find and buy online, if not available locally, is a better substitute (as a descendant of the original garum, perhaps this is to be expected).

References:

Athenaeus, Deipnosophistae: http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Athenaeus/2C*.html (see 67C)

Sally Grainger, "Roman Fish Sauce: an Experiment in Archaeology", pp 121-131 in Helen Saberi (ed), Cured, Smoked, and Fermented: Proceedings of the Oxford Symposium on Food and Cooking, 2010, Oxford Symposium, 2011

Liji, Neize: https://ctext.org/liji/nei-ze (see 33)

Ranendra K. Majumdar, Deepayan Roy, Sandeep Bejjanki, Narayan Bhaskar, "An overview of some ethnic fermented fish products of the Eastern Himalayan region of India", Journal of Ethnic Foods 3(4), 276-283 (2016) [open access]

Magnus Nilsson, The Nordic Cook Book, Phaidon Press, 2017

Namrata Thapa, "Ethnic fermented and preserved fish products of India and Nepal", Journal of Ethnic Foods 3(1), 69-77 (2016) [open access]