Let $f$ be the density of the stock asset under some model (Heston, SABR, Black Scholes, Variance-Gamma, etc).
Is $f$ square-integrable in these models?
I've never heard about the density function of the stock price in Heston model. For pricing, one uses the characteristic function which can be derived from the Heston-PDE. In general, a density function is not always square-integrable. Check this post https://math.stackexchange.com/questions/756540/is-a-probability-density-function-necessarily-a-l2-function
Yes, $f$ is square-integrate. This follows from the fact that the variance exists.
Note that in general for most models this is the case but it is debatable given that financial returns are so heavy tailed. Thus, by modeling finance with models that output returns with finite variance we might be heavily underestimating the risks.