This is an interesting idea, but I don't think it's all that practical - or at least not as good as you seem it to be. Still, you can try it, just don't expect wonders.
At what temperature would this type of silk start to decompose?
Strictly speaking, at about 40 to 45 degrees Celsius. Silk is, after all, made out of proteins. This is why silk clothing, if washable at all, is labelled for 30 degrees wash.
I said "strictly speaking", because it won't decompose the way a paper towel decomposes in the washing machine. It will change its texture, but not fall apart. It will get more brittle each time you use it - but you will get multiple uses out of it, unlike a paper bag.
Out of the silk types you listed, I would go with organza. There are commercial (synthetic fibre) teabags which mimic organza, and they work well. It is permeable enough to be a good filter, and quite form-stable. A second choice would be chiffon, and maybe etamine if you can ensure only large plant parts. Habotai/ponge of the thinner kind would also be permeable, but it's so fine I suspect it won't last too long. Other kinds (twill, georgette, velvet) are decidedly too dense, and also quite expensive.
You will have to pay lots of attention to the construction. Silk is very delicate, and frays easily. And since you will be cooking it frequently, and possibly exposing them to sticks and berries and hard leaves, you can expect damage to emerge soon. You'll need an overlock, and I wouldn't just seam-and-serge in one, but really do overlock on the exposed sides, and additionally do the side seam, maybe even a jeans-strength seam.
And a last note, while cotton is indeed very absorbent, I would still expect flavor to carry over between brews with any natural fibre you choose. Silk may be less prone to it, but not zero. And pay attention to what kind of tea you are making in it, because silk is more reactive to alkalis than to acids, so some herbs will damage it more quickly than others. Keep it out of direct sunlight, even for drying it, since it's rather sensitive to UV. And finally, pay attention to signs of mold, it can happen during storage, and your bags, unlike clothing, will be frequently wet.
Added thoughts: With all these caveats above, you might after all look into fabrics that are not pure silk, to reduce the speed of disintegration. Ideally, you'll accept some amount of synthetic fibres, since they are what can make a fabric strong. If not, maybe look into something like a silk-flax voile, if you can find it. But since not all fabrics of the same content are created equal - a silk thread around a synthetic core will create a many times stronger fabric than a silk thread shot with a bit of lurex for sparkles, even if both say "90% silk, 10% nylon" - it is best to go to a specialized store and get advice in person. Also, very important - touch the fabric, evaluate how dense it is, how stable, and whether you have the skills to sew it (silk is notoriously difficult and requires a good sewer and a good machine). This means that you shouldn't go to the corner fabric shop that caters to quilters and organic-jersey-moms, but to somebody who specializes in rare natural fabrics, for example for theater supply or historic reenactment.
