How can I use the Heston Model to calculate the probability of a stock being above or below a certain value on a given date in the future?
In options pricing language, the probability of a spot process being above a given level $K$ at time $T$ is the undiscounted price of a digital call option on that spot process. In the Heston model, there is an analytic expression for this in terms of Fourier transform. You can find this in various standard references, e.g. Alan Lewis's book "Option Valuation Under Stochastic Volatility" or by google search. When I try "digital option in heston model" the top result I find is a student paper http://www.cs.ubbcluj.ro/~studia-m/2003-3/lazar.pdf which at first glance looks basically correct, though I don't see a discussion of the branch cut issue which often trips up unwary implementors of Heston model analytics.
COS method is an efficient way to recover the distribution function from the characteristic function in the Heston model. For other methods, you may refer to "Inverting Analytic Characteristic Functions and Financial Applications".