The only source for the Immortals is Herodotos' account of Xerxes' invasion of Greece. To my knowledge, this is the full extent of what we know about the Immortals:

After that, first came a thousand horsemen, chosen out of all Persians; next, a thousand spearmen, picked men like the others, carrying their spears reversed. (...) Behind Xerxes came a thousand spearmen of the best and noblest blood of Persia, carrying their spears in the customary manner; after them a thousand picked Persian horsemen, and after the horse ten thousand that were foot soldiers, chosen out of the rest of the Persians. One thousand of these had golden pomegranates on their spear-shafts instead of a spike, and surrounded the rest; the nine thousand who were inside them had silver pomegranates. Those who held their spears reversed also carried golden pomegranates, and those following nearest to Xerxes had apples of gold.

-- Hdt. 7.40.2, 41

These were the generals of the whole infantry, except the Ten Thousand. Hydarnes son of Hydarnes was general of these picked ten thousand Persians, who were called Immortals (ἀθάνατοι) for this reason: when any one of them was forced to fall out of the number by death or sickness, another was chosen so that they were never more or fewer than ten thousand.

-- Hdt. 7.83.1-2

When the Medes had been roughly handled, they retired, and the Persians whom the king called Immortals, led by Hydarnes, attacked in turn. It was thought that they would easily accomplish the task. When they joined battle with the Greeks, they fared neither better nor worse than the Median army, since they used shorter spears than the Greeks and could not use their numbers fighting in a narrow space.

-- Hdt. 7.211.1-2

When they had arrived in Thessaly, Mardonios first chose all the Persians called Immortals, save only Hydarnes their general, who said that he would not quit the king's person (...) The Persians whom he chose (men who wore torques and bracelets) were more in number than those of any other nation and next to them the Medes; these indeed were as many as the Persians, but not such stout fighters.

-- Hdt. 8.113.2-3

Where Mardonios was himself, riding a white horse in the battle and surrounded by a thousand picked men who were the flower of the Persians, there they pressed their adversaries hardest. So long as Mardonios was alive the Persians stood their ground and defended themselves, overthrowing many Spartans. When, however, Mardonios was killed, and his guards, who were the strongest part of the army, had also fallen, then the rest too yielded and gave ground before the men of Lakedaimon.

-- Hdt. 9.63.1-2

It should be obvious that, with this source base, some of your questions cannot be answered. No Persian records exist to complement Herodotos' Greek perspective. There are various later sources which have been assumed to describe or relate to the Immortals, but none of them ever use the name, and so much of this is essentially speculation.

The biggest frustration is that we know nothing about this history of the unit before or after Xerxes' campaign. Who raised it, and when? How were its men selected? Where was it garissoned? How long was it maintained? All we can say is that if there really was such a royal guard (and we have no reason to doubt Herodotos on this point), it should follow that this guard is the one depicted on the palace reliefs at Persepolis, and that this guard would have been present at later engagements such as Kounaxa and Gaugamela. But we do not hear about it in the surviving accounts of these battles. It is generally assumed that the Immortals formed the professional core around which Persian field armies were built - but we have nothing but Herodotos to go on, and no way to test whether he was telling the truth.