How does the Grail choose the participants?

Question

How does the Grail choose the participants?

• We know that the Grail tries to pick people who have some sort of strong desire (when Kotomine asks Tohsaka at the beginning of Fate/Zero).
• Do you need to descend from a mage bloodline? Did the murderer guy (the one who got Caster in Fate/Zero) have magical blood?
• It seems like you don't need to be aware of the existence of the Holy Grail war (Shirou and the murderer guy were both pretty clueless).
• You do seem to need an heroic relic... or do you? When Kotomine was surprisingly chosen in Fate/Zero, did he have one already? What if you lose the relic before the summoning occurs? (In Fate/Zero the participants were already making preparations for years before the summoning).
• The Matou were disappointed that Kariya left behind their magical teachings, so they decided to adopt Sakura to prepare her for the next war. When Kariya returns and chooses to fight, he successfully becomes a master. This suggests that there is a way to opt out/in from the war (?)

So how does the Grail pick participants exactly?

Answer 1

The exact way in which the Grail picks Masters is not well-understood, as far as I'm aware. It's a fuzzy process even to the participants themselves.

To try to address your specific questions:

Do you need to descend from a mage bloodline? Did the murderer guy (the one who got Caster in Fate/Zero) have magical blood?

I'm not sure whether you strictly need to descend from a mage bloodline (this is probably addressed in side materials I haven't read), but Ryuunosuke did have a mage somewhere in his ancestry, and as such, he had a small number of Magic Circuits (this is mentioned in the light novel, but not the anime). The magic tome he had on hand when he murdered that one family right before summoning Caster was apparently something he found in a storehouse belonging to someone in his family.

It seems like you don't need to be aware of the existence of the Holy Grail war (Shirou and the murderer guy were both pretty clueless).

This is true.

You do seem to need an heroic relic... or do you? When Kotomine was surprisingly chosen in Fate/Zero, did he have one already? What if you lose the relic before the summoning occurs?

You don't need a relic to be granted Command Spells by the Grail. Having a relic helps with the summoning further down the line, but even then, you don't strictly need a relic. If you don't have a relic, the Grail just gives you a Servant who is "compatible" with you or "similar to" you in some sense.

The Matou were disappointed that Kariya left behind their magical teachings, so they decided to adopt Sakura to prepare her for the next war. When Kariya returns and chooses to fight, he successfully becomes a master. This suggests that there is a way to opt out/in from the war (?)

Kariya had no guarantee that he would be selected as a Master during the fourth war. He was essentially gambling, hoping that by undergoing the worm treatment, he would be picked by the Grail.

Practically, I get the impression that there are usually few enough mages in the Fuyuki area that most mages in town during a Grail War end up becoming Masters.