EDIT: I update while making progress:
I am trying to build (model implied) volatility surfaces for individual equities. I will use these surfaces to calibrate models to price different derivatives (plain vanillas and exotics). An important note is that I intend to use these for sensitivity computations and not for pricing efficiency. I would like to validate my methodology from a theoretical and a practical perspective and ask several questions:
Instruments: American options are the most liquidly traded intruments and the only ones that I can think about for this purpose, so I use these points for this exercise.
Model implied volatilities: use Cox-Rubinstien-Ross (CRR) binomial tree to price American options by including discrete dividends from forecasted dividends. Use root finding algorithm to find the Black-Scholes (BS) implied volatility. This is done by recomputing the option price by using the CRR tree for each candidate implied volatility parameter proposed by the root finding alogirithm.
Fitting method for the implied volatility surface: Here I refer to the great answer by Christian Fries What are the advantages/disadvantages of these approaches to deal with volatility surface? I go with the SVI/SSVI approach with techniques described in https://arxiv.org/pdf/1204.0646.pdf to avoid calendar spread arbitrage. I choose this method because it allows me to both be accurate enough for plain vanilla option valuation, it also give a good representation of forward volatilities for other derivatives.
Pricing model calibration: I then use the obtained surface to get European option prices by applying the BS formula. I use these European option prices to calibrate a pricing model, e.g. Heston, local volatility, by minimizing these prices with the model-derived European option prices.
Should I build two surfaces, one for puts and one for calls?
Is it possible to have a single equity implied volatility surface to use for pricing different derivatives? Should I use the same model as my pricing model to build the volatility surface?
Is there any good literature that could clarify these practical issues? I am confused because topics of model calibration and surface fitting are usually treated as two separate things. Are the fitting methods only needed to extrapolate the prices of plain vanillas in an arbitrage-free way? How is it done in practice?
Do the same principles apply for FX, and roughly to interest rates? In the interest rate context, my understanding is that it is done in two steps 1. build a model implied volatility surface by e.g. inverting Black's formula and calibrating SABR. Then for calibration of a model (e.g. Hull White) use the derived e.g. swaption prices from this surface and match them with the pricing model derived swaption prices (from Hull White in this case).