Here's the problem--the natural rubber content in the Kazakh dandelion is about 8-10x lower than modern rubber trees. I am not certain how much less it is than Castilla elastica, the rubber source used in Mesoamerica, but it probably is quite a bit less since after all, nobody used these sources in ancient times despite growing in areas directly on the Silk Road with ancient urban civilisations. Apparently the process involves refrigerating the roots and then grinding them, which using modern machinery gets about 90% of the rubber from the root. It was less in World War II, and no doubt pre-industrial civilisation would achieve far less plus would have the added problem of only being able to harvest the plant on certain cold days and using priceless ice to refrigerate it (or long trips into the high mountains).
So assuming it's even possible, Old World rubber and its products would remain a novelty item strictly for the elite probably across the entire Silk Road, even if they did find a way to vulcanise it (which might not be too hard--in Mesoamerica the rubber was mixed with the latex of another plant of genus Ipomaea which happened to have a lot of sulfur compounds in it, so such a technique might be doable).
I think pretty obviously it is sadly hard to do anything with the known OTL properties of even the Kazakh dandelion. (And silly me, here I was believing that just about any kind of dandelion would work as well as any other...)
This is not ASB forum so we can't just postulate another natural variety more suitable for some reason to yield a decent amount of decent latex without ice, refrigeration, etc.
It would not be ASB however if there were a plausible path to ATL cultivation of this dandelion, or some relative not very distinguished OTL that happens to have a plausible path, over thousands of years of cultivation, to become more useful in the manner the vast majority of modern crops are radically different in obvious ways from their wild ancestors. If someone way back in the early Neolithic, in Central Asia or anywhere else--say this happens in what would become Gaul, completely ATL but possibly not ASB--cultivated this hypothetical Gaulish Latex-Dandelion probably for some other reason than developing latex, but found themselves processing latex as an undesired waste product perhaps as a side effect of a desirable feature--say they eat the dandelion greens, and the root if processed to get the latex out is also of dietary worth and much more so in the cultivar variant, but somehow the selection for better greens and roots paradoxically also both increases latex production, in healthy well tended cultivar plants anyway, and largely sequesters it, say in the central part of the taproot, while the periphery is where the food value is--or vice versa, a latex-infused sheath over a good food core perhaps. Anyway, after a few thousand years of flensing the gummy outer sheath off the edible core and then just tossing the sheath aside, people fool around with it and discover some random eclectic uses for it and broaden their repertoire; they learn to efficiently separate the latex from its tough outer root matrix, and treat it various ways, one way makes it very sticky in treated parts while being a firm grip on untreated parts of a rod or block, other ways make it closer to what we moderns consider proper rubber. The bouncy aspects can indeed have some uses perhaps in cart wheel axle suspensions.
Aside from where it develops, Gaul or Central Asia or Iberia or North Africa perhaps, I think if we can justify a non-ASB progression like this in the dandelion's heartland as a cultivar (quite possibly coterminous with the natural habitat range, in the Interglacial, of the wild ancestor strain) we also ought to move the "POD" as far as ATL Hellenistic people messing around across the Mediterranean with it, or rather starting to, back some centuries. Say 3, and it is more like 150 BCE rather than CE--so basically in the period when the Roman Republic was fast approaching the crises that reworked it into the Empire that the plant starts being of some interest to the Hellenistic upper classes, perhaps also involving the possible uses as food for the lower classes, a la the rationale behind the RN sending the Bounty under Captain Bligh to acquire breadfruit cuttings and seeds from Tahiti to import to the West Indies as food for slaves. So now we have both patricians and the lowliest commoners fooling around with goofy mealroot sheaths and the arts already known in its heartland to get some uses out of them largely spread with the plant itself. Then, over the next three hundred years or so, first the dandelion is spread around eclectically, being introduced here and there mainly for its food value, where it has very hit or miss success. There are plenty of places where the agriculture is getting marginal due to a couple thousand years of intensive cultivation in the Med, and in some of those places the weed-variety does relatively a lot better than more normal crops, and so there is some tendency for marginal people to grow it on the side, especially if it doesn't really require a lot of care just to grow--to make it really a good food crop, you have to lavish a lot more care on it, but if you just give it some rather light attention in a spot marginal to useless for normal crops, it grows fairly well if not in the most delectible form, and it provides nourishment supplement for remarkably low intensity or time span of labor attention given it. It does better if well weeded of course but it can tolerate quite a few weeds intruding, tending to out-weed them. It proves to be a useful companion plant to something or other; maybe helps improve the soil in ways such as suggested upthread on more limited forms of the lines I am suggesting.
And then, with the stuff quasi-ubiquitous (that is, you travel around among many versions of essentially the same climate zones, and in some you find the dandelion is either running wild or under cultivation, and in others it didn't take, for reasons future agronomists will find themselves turning to purely social history as well to fully explain) then some alchemist-tinkerers, or perhaps artisans goofing around with the wacky tacky stuff, stumble upon forms approximating vulcanization and other possible treatments, and the utilities become much improved--say they can develop a form one can make fairly strong Bungee cord type arrangements, something that with enough short lengths of it in parallel one might greatly damp out the ride of carts and chariots for instance. (No one uses chariots in combat any more, but they are ceremonial). By now it is pretty late into the Roman empire, right in one of the centuries that seemed a golden age in retrospect, so useful ideas are extra likely to spread versus earlier or later times.
This could shift the basis of cultivating the dandelion away from its food value toward its latex value--but perhaps breeding the plant for better and more latex at this point again paradoxically leaves it actually almost as good a food plant, so actually everyone grows it for both. Meanwhile the latex types are clading out into distinct varieties and some of them have notably different latex properties than others.
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I actually have remarkably little interest in the rice subject, it seems pretty well done by others.
The rubber is very much a long shot, but I don't think it is proven to be ASB, though I do think starting the clock with an introduction in 150 CE instead of three or more centuries before is a way of moving the eventual impacts well out of the Roman Era into the Middle Ages sometime.
Also it does matter where the home zone of the ATL favored variety was. After suggesting Gaul pretty much at random, I thought maybe North Africa might be a more sensible place, say as a weed on the borderland between semi-arid but cultivatable land and the dryer edge of the desert proper. So if it comes from Gaul it is more and more a European thing, with varieties gradually adapting somewhat to more continental climates and shifting eastward essentially with Latin Christendom, which develops it ubiquitiously in Europe through the Middle Ages. While if it is Tunisian or Moroccan, I suppose after North Africa goes Islamic, it becomes very much an Arab/Muslim sphere sort of thing not so well known beyond the Mediterranean itself.
Anyway to make it a Roman Imperial age thing I think we need to introduce it a lot earlier, and withal with some thousands of years of ATL cultivation behind it before then.
