I don't have much to add, but wanted to address the "price of risk" question.
APT is kind of "economics"-free and tries to price assets without the utility maximization required in CAPM/ICAPM. Ross's APT observes that groups of assets move together (e.g., tech stocks) and that is the risk you're bearing because the idiosyncratic risk, like the firing of HP's CEO, can be diversified away. Because this risk is easily diversifiable, the market won't pay you to take it. So in your APT model these factors are returns to asset classes, industries, etc.
Although the model looks the same, in Merton's ICAPM, the factors are state variables (e.g., industrial production, inflation). These are purely academic points -- in practice you run a multivariate regression with return on the LHS and whatever factors you think are priced on the RHS. OLS and GMM are common. So you'll estimate $$E ~ \left[ ~r_i~ \right] = \alpha_i + \beta_i^1 f_1 + \beta_i^2 f_2 + \ldots + \beta_i^k f_k$$
Your final question.
Also how is the return attributable to
a specific factor calculated?
Now you regress the returns back on the betas. $$E ~ \left[ ~r_i~ \right] = \sum_{j \in K} \lambda_j \beta_i^j$$
Where $\lambda_j$ is the return to factor $j$. Typically the Fama-MacBeth approach is used here. If you've done it correct and found something, $\lambda > 0$ (i.e., the market is paying you to take this risk).