There is no one answer to this question.
To the first part of your qurstion, yes demand plays a factor in determining the pricing but not entirely, there are other factors as well like in monopoly or oligopoly Marginal costs of the firms play an important role as well. But when it comes to perfect competition demand does not play any role in determing the price of the market as it is totally determined by the marginal cost of the competing firms which is assumed to be same in perfect competition.
Now if you leave out price discrimination done of the monopolist, the answer to the second part of your question is yes. Price determines who gets what. But if there is price discrimination then monopolists decides which market gets what product at what price.
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