Did Manjushri, Vajrassatva, Avalokiteshvara live?

Simply, in Mahayana Buddhism, they exist. All three of them are or were Bodhisattvas and are or will become fully Enlightened Buddhas upon the completion of their vows. The qualities of a tenth ground Bodhisattva are pretty close to the qualities of a fully enlightened Buddha.

Now for the complications:

They are not historical personages in the same way that, say Gautama Shakyamuni was. They are either myths, i.e. instructive fictions, or if one accepts the possibility of realms and other universes that can be known through meditative experience, then they are the Buddhas of other universes. A common "rule" in discussion of the realms is that there can only be one Buddha per universe at one time, with colossal gaps of time between them. So either these Buddhas are in other worlds (in one of the ten directions) or they existed a very, very long time ago. A universe can have many Bodhisattvas and Buddhas and Bodhisattvas can visit other universes through the use of other bodies (nirmanakaya).

Existence, to me, is not so important. If they exist, they are anatman, i.e. they don't have an unchanging core, they are empty/sunyata. I don't need them to exist with a birth certificate and the like.

Depending on who you are reading, you get different accounts of who is a Bodhisattva (not fully Enlightened) and who is a fully Enlightened Buddha. This confusion is compound when you take into account how the Bodhisattva path is explained-- a Bodhisattva reaches the edge of Enlightenment, but holds back because there are still more unenlightened beings in the universe. By this reasoning, you personally will be a fully enlightened Buddha before the Bodhisattvas are Buddhas.

Also, in the explanation of what Enlightenment is, the distinction between unenlightened and enlightened might not be what we think it is. In my readings the impression I get is that if we were nearly enlightened, it would all be clear. But as unenlightened beings, the distinction between the Bodhisattva and Buddhas are more distinct than what we naively think, in fact it may be that samsara and nirvana are in some sense the same and that the Bodhisattvas and Buddhas were always the same.

Vajrasattva did not live in the sense you mean, because he is not a person, but he is an existent. I will explain further:

The form body (Rupakaya), is divided in two:

1. The emanation body (Nirmanakaya), and
2. The enjoyment body (Sambhogakāya)

From a Tibetan viewpoint, Shakyamuni was an emanation body of the Buddha. He merely showed the aspect of being born, aging, achieving enlightenment, passing into parinirvana, and so forth [but was actually free from birth, aging, sickness and death and so forth]. While the enjoyment body is too subtle for ordinary beings to see, the emanation body is seen by ordinary beings. Wheel-turning buddhas are thus said to be emanation bodies.

Vajrasattva, on the other hand, would be the enjoyment body in the aspect of a Bodhisattva. Such a body can be seen by arya beings. We claim that when an arya bodhisattva chooses to be reborn in a pure land in order to receive teachings there from other bodhisattvas, he receives teachings from actual bodhisattvas as well as from the enjoyement body in the aspect of bodhisattvas (such as Vajrasattva, Manjushri, and so forth). Vajrasattva is one of these aspects, manifestations. He is a subtle manifestation of the form body of buddha.

Avalokiteshvara was one of 'the eight principal Mahayana disciples of Buddha Shakyamuni', that is the sambhogakaya in the aspect of a disciple. The same is true of Manjushri. The main difference between Manjushri and Vajrasattva, in relation to their entity, is that Manjushri is said to be a [manifestation of the] wisdom truth body (i.e. the omniscient mind) and not that of the enjoyment body.