I checked out a paper which deals with out-of-sample option pricing (http://repec.kse.org.ua/pdf/KSE_dp38.pdf, especially following pp. 40-) and I believe it is a sound approach to test whether the addition of structural parameters ads value in pricing capability to more parsimonious models.
Their approach is to
- derive additional parameters (I use the term parameter as in parameterized model, additional in the hopes of obtaining a better fit between the model outputs and actual prices), hoping to derive a model that results in a better fit.
- However, the danger is to overfit such models and they use "out-of-sample" tests in order to verify whether the improvement in fit is merely a function of overfitting or whether the additional parameters display true added forecasting value.
- The structural parameters are fixed to t but all other option pricing model inputs are varied over time when calculating option prices at t+1, t+2,...,t+n, hence they speak of out-of-sample testing of structural parameters.
- The authors' conclusion of this particular paper is that indeed the approach, that alternative models take by inclusion of structural fitted parameters, is a viable way to model option prices aside the general stochastic volatility models. They state that out-of-sample tests have shown that such parameterized models are able to correctly capture the volatility term structure and smile.
In summary, I would answer question1 with a "reserved" yes if you mean with out-of-sample the testing of structure parameters rather than all model inputs (as described above other time varying parameters, such as stock price,..., are not out-of-sample). The second question I would answer in that the out-of-sample structural parameters at t are used to estimate option prices at (t+1,...t+n), and therefore the other parameters are kept varying by t to zero in on the structural parameter fitness, not on the other parameters.
So, my hunch is that the papers you came across focus on the structural parameters, only, when they mean "out-of-sample testing", while you maybe thought that out-of-sample testing includes all model inputs and thus were perplexed why stock prices at t+1...t+n are used to estime option prices over the same t+1...t+n.