Are there any descriptions of the deathless/unborn in any tradition of Buddhism?

I think that in the Pali suttas:

• "The deathless" is a synonym for nibanna (see this answer).
• Samsara is conditioned and has no construable beginning (which, I think, means 'perpetual' but 'without an original creator')
• Nibanna is (in contrast) unconditioned

I don't think there is (I don't know of any) doctrine which explains how/why conditioned existence comes into being from the unborn.

There is a theory of dependent origination which explains how one thing leads to another, i.e. how samsara is perpetuated.

And I think that Buddhism is seen as a way to escape from (to be liberated from) samsara.

Speaking of Siva, I've noticed that akalika (timeless) is given as one of the attributes of Dharma. Kali is a (Hindu) goddess of Time, consort of Siva, and associated with (among other things) Destruction.

So, conversely, perhaps it makes sense that that which is akalika is associated with what's Deathless (and Unborn).

My understanding of it in conventional terms is that "Born" means that 'it' comes into existence as a result of conditions, and Death means that it stops existing in the same way (see e.g. this essay, Anicca Vata Sankhara by Bhikkhu Bodhi) -- and thus, 'it' is impermanent.

Conversely the Unborn and the Deathless is not impermanent.

In its description of the Mahāparinirvāṇasūtra, The Princeton Dictionary of Buddhism claims,

Th nirvana of the arhat is said to be merely the absence of the afflictions (KLESA) but with no awareness of the buddhadhatu. The nirvana of the Buddha is instead eternal, pure, blissful, and endowed with self, a primordially existent reality that is only temporarily obscure by the klesa; when that nirvana and buddhadhattu are finally "recognised," buddhahood is then achieved. The Buddha reveals the existence of this nirvana to boddhisatvas.

Incidentally it also says,

To assert there is no self is to misunderstand the true dhamma. The doctrine of emptiness (SUNYATA) this comes to mean the absence of that which is compounded, suffering, impermanent.

So if you're looking for doctrine about the "unborn" as something other than an absence of defilement, something with a self-existence, then you might want to be looking into later Buddhist doctrines, for example to do with Buddha-nature.

If you read the Mahāparinirvāṇasūtra there's a lot of talk about Self, for example,

Kasyapa said to the Buddha: "O World-Honoured One! Is there Self in the 25 existences or not?" The Buddha said: "O good man! "Self" means "Tathagatagarbha" [Buddha-Womb, Buddha-Embryo, Buddha-Nature]. Every being has Buddha-Nature. This is the Self. Such Self has, from the very beginning, been under cover of innumerable defilements. That is why man cannot see it.

You're asking "what it is", which I think assumes that "it is": i.e. that there is an 'it', which is existent. That existent 'it' I suppose might be Buddha-nature (which I think the Pali sutta describe more as a non-it: not-born, not-dying, trackless, unconditioned, not suffering, etc.).

Even so I'm not sure that it's right to talk about the unborn as a creator or source of conditioned existence.

Apart from "Buddha-nature" there's another word from the later doctrine, Tathāgatagarbha,

The Tathāgatagarbha sūtras are a group of Mahayana sutras that present the concept of the "womb" or "embryo" (garbha) of the tathāgata, the buddha.

Calling it "womb" implies that maybe it's a creator of something, but I think that what it's said to be creating is Buddhahood rather than creating conditioned existence:

Every sentient being has the possibility to attain Buddhahood because of the tathāgatagarbha.

For example the Tathagata-garbha Sutra says,

"Or, kulaputras, it is like a kernel of wheat that has not yet had its husk removed. Someone who is impoverished might foolishly disdain it, and consider it to be something that should be discarded. But when it is cleaned, the kernel can always be used. In like fashion, good sons, when I observe beings with my Buddha cakshur, I see that the husk of kleshas covers their limitless Tathagata vision. So with appropriate upayas I expound the Dharma, to enable them to remove those kleshas, purify their jnana paramita (tenth bodhisattva stage) and to attain in all worlds the anuttara-samyak-sambodhi."

... and,

"Or, kulaputras, it is like the genuine gold that has fallen into a pit of waste and been submerged and not seen for years. The pure gold does not decay, yet no one knows that it is there. But suppose there came along someone with supernatural vision, who told people, 'Within the impure waste there is a genuine gold trinket. You should get it out and do with it as you please.' Similarly, kulaputras, the impure waste is your innumerable klesha. The genuine gold trinket is your tathagatagarbha. For this reason, the Tathagata widely expounds the Dharma to enable all beings to destroy their kleshas, attain correct perfect enlightment and perform Buddha deeds."

In summary I don't know of a doctrine which says that the conditioned comes into existence because of the unborn.

If anything perhaps it's the reverse, e.g. there's the image of the (pure) lotus which grows out of the (impure) mud.

Even so, attaining the unborn might change the way in which you see conditioned existence.