What's a "correct price" really? Does it exist?

Your question is actually 3 or 4 semi-independent ones, but I'll take your cue and attempt to answer in the same manner.

Adam Smith in his book "Wealth of Nations", Book 1, argues that there are two distinct forms of value in "every thing":

1. Value of use; the value one has from using said thing.
2. Value of exchange; the amount of another thing that said thing can be exchanged for.*

When exchanging a product for another, you get the real price of both things, in a relative fashion. This relative price of both things depends from how each person values the product they give in relation to the one they take.

When you exchange a product for money, you get the nominal price of said product. This nominal price corresponds to the real price in a mutually agreeable fashion for both parties of an exchange; meaning that if money has an intrinsic value of its own, mostly value of type (1), then fluctuations of that value will change the nominal price that each good can be exchanged with money for. However the real price of this product hasn't changed, ceteris paribus.

*Smith speculates that labor is the factor with which value of exchange is ultimately decided independently by each party, though this decision is mostly arbitrary - it is not precisely quantifiable before setting the price.

Notes:

I would suggest you look into Walrasian equilibrium for a methodical theory price setting, and of course, please do read Adam Smith's Wealth of Nations; the first book provides theories for most of your questions.

tl;dr as KennyLJ correctly points out the concept of correct price is meaningless and it does not exist or rather you could say any price is the correct price (but again this just makes the whole concept meaningless because if every price is correct then thats just a price - a concept of 'correctness' makes only sense if there are some incorrect instances).

The reason for this is that there is no 'intrinsic' objective value to anything. Since all value is completely subjective without even slightest objective component there cannot be any correct or incorrect price because there is no correct or incorrect value to anything, every valuation and any price is correct and hence the word 'correct' loses its meaning.

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Full Answer:

The question of what is the 'correct' price is actually interesting from the perspective of history of economic thought and it actually dominated economic thought in ancient and medieval times and in some non-economic thinking it persists even to this day. In those times what is correct was heavily influenced by ethical considerations as at that point economics did not yet emerged from moral philosophy and could not be divorced from all other subject subsumed there (similarly as all natural subjects were at the time part of natural philosophy). As a result the early economic thinking was dominated by the concept of 'just price' which was considered both moral and desirable. Furthermore, just price and any other thinking on 'correct prices' relies on notion of objective intrinsic value.

Paul Mueller & Jan Gerber discuss the history of 'just price' concept in great detail I will do it here only briefly. The concept already originated with Aristotle in his Nicomachean Ethics and Politics. In Aristotelian thinking a just price would be a price at which the same value is exchanged. That is a just price wold be a price at which $$1$ of value embedded in apples exchanges for exactly $$1$ value embedded in peaches.

Thomas Aquinas expands on the Aristotelian concept according to Muller & Gerber by using the following innovations:

While it is still wrong to sell something for more than it is worth, it is not wrong to sell something for more than one paid for it. While Aristotle saw injustice in the shoe example, Aquinas recognized that one can do something productive or useful with a good that improves it in such a way that its value is higher, and so a higher price is still just. Transporting the shoes from one town to another represents a kind of improvement, as does any kind of labor performed on the good.

While this is a little more sophisticated than Aristotle’s account, it still relies on an “intrinsic” or “natural” value of goods—with a focus on the costs or input which creates value. Furthermore, Aquinas and other scholastics were also wary of profit. Some profit, a “reasonable” amount, was fine. But excessive profit could only be the result of injustice and exploitation.

Later when economics started to emerge as a separate field from moral philosophy classical economists started to turn away from the moral notion's surrounded around the 'just price' (since in science we like to make sharp distinction between normative value judgement and positive inquiry) although as the article by Muller & Gerber explains in moral philosophy the notion lived on for a bit more, but the classical economists such as Adam Smith or David Ricardo did not turned away from the notion of intrinsic value which would still enable a possibility for 'the correct' price to exist.

In fact Adam Smith was firmly behind the idea that there is some intrinsic value to goods and services. As Steven Horwitz writes:

Smith is very clear in The Wealth of Nations that he sees labor as the source of value. For example, in the opening paragraph of Chapter 5 on real and nominal price (I.v.1., p. 47), he writes:

The value of any commodity, therefore, to the person who possesses it, and who means not to use or consume it himself, but to exchange it for other commodities, is equal to the quantity of labor which it enables him to purchase or command. Labor, therefore, is the real measure of the exchangeable value of all commodities.

And later in that chapter (I.v.7., p. 51):

Labor alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price; money is their nominal price.

However, as our scientific understanding progressed any notion of objective intrinsic value got completely shattered. This occurred during the marginal revolution where subjective theory of value was virtually simultaneously discovered by Jevons, Walras, and Menger in the late 19th century.

The subjective theory of value claims that there is no objective intrinsic component to any value and all value is completely subjective given by person's own preferences. This theory of value can explain situations that labor theory of value or other theories of value cant. For example, under labor theory of value digging hole and immediately covering it up should have some intrinsic value due to labor component embedded in it but we can observe that people dont value such absurd waste of labor. This can be fully explained by subjective theory of value that would simply postulate that the dig up and immediately covered hole is worthless because nobody values it (or if someone actually pays for it it has some subjective value of them). Conversely, even things that do not have any labor embedded in them can be valuable. Furthermore, value further fluctuates based on what are people's marginal utilities. As the quotes mentioned in the_rainbox's answer from Adam Smith show, classical economists already argued that individuals have different levels of utility or 'value in use', but did not properly connect them to prices, or 'value in exchange', considering them separate, one derived from the quantity of labour input and other production factors (see Stigler's The Development of Utility Theory. I).

Currently subjective theory of value dominates the mainstream economic thinking, and does so to the point that with perhaps a little bit of a hyperbole the detractors from this theory are in similar number and position as anti-vaxxers in medical science. You will find the subjective theory of value in every modern economic textbook and arguably it is either implicit or explicit in wast majority of the whole corpus of economic literature written post marginal revolution. What even more, the subjective theory of value 'spilled' from economics back to the moral philosophy where it was used to nail the last nails into the coffin of the moral theory around 'just price' (see the article by Mueller and Gerber again).

As prefaced at the begging under subjective theory of value, objective intrinsic value that could be used as a justification for correct price simply does not exist. If two people make any voluntary transaction in the market they must either exchange equal or both greater value than was lost (or some combination of thereof) meaning if any transaction occurs the price is automatically the 'correct' price for people engaging in that transaction. Furthermore, even if a transaction does not occur because maybe seller asks for too large price or buyer for too low price you cannot dismiss their subjective valuations as incorrect and the prices they are asking for as an incorrect prices. Once you adopt subjective theory of value any notion of correct price becomes completely absurd. Hence the answer to your question is that there simply isnt any 'correct' price.