How does US preclearance work in Dublin?

"like a citizen" should read "as if I were arriving from another US airport" because citizenship has nothing to do with whether a passenger must clear immigration and customs. Only the point of origin is relevant.

You cannot use preclearance if you're flying through Amsterdam because this statement is incomplete:

To go to preclearance, I have to have an already purchased ticket to LAX from Dublin.

It should actually say

To go to preclearance, I have to have an already purchased direct ticket to the United States from Dublin.

The problems you're asking about do not exist because preclearance applies only to direct flights. Either all the people and goods on the plane have been cleared, or none of them have. But your questions are a good illustration of why preclearance is only allowed on direct flights:

If I am precleared to enter the US (without an immigration check at LAX upon arrival), how will they know that I have been precleared, and they have to treat me differently from other passengers who are not precleared?

There's no way for them to know that, which is why the entire plane must be precleared, or none of it. Some precleared planes even land in airports like New York's La Guardia, where CBP does not process international commercial flights. The first awareness I ever had that preclearance exists came when I noticed an Air Canada flight at La Guardia.

Upon arrival at LAX, there are two kinds of lanes, one for US citizens (and permanent residents), and another lane for non-permanent residents. Now normally I would go to the non-permanent resident lane and would have to go through CBP check. But in this case being pre-cleared, will i be able to enter the citizen lane?

This points out another inconsistency in your thinking: a pre-cleared foreigner is not in the same category as a non-cleared US citizen. The latter must present a passport to a passport inspector, while the former has already done so.

Other answers are correct, but to be concise: Preclearance is not a service that passengers may choose to use, by having purchased a ticket to the United States. Rather, pre-clearance is part of the pre-boarding procedure, precisely for passengers boarding direct flights to the United States from designated pre-clearance facilities. Since all people on such flights (and their baggage) will have been admitted to the United States before boarding, they are generally not subject to border inspection at all when arriving in the United States - the flight is treated as if it is arriving from within the United States.

Airports with Preclearance facilities are listed here, and the vast majority of direct flights from those airports to the United States will use the preclearance facilities. However, even from a departure airport with preclearance facilities, preclearance may sometimes not be employed; passengers on a flight not precleared will then pass through immigration and customs upon arrival in the US.