Non stationary model applied to stationary data If the model is non stationary, can it be applied to stationary data/series?
For example the KPSS test shows that the data is stationary but the BP test shows presence of heteroscedasticity. 
If a stationary model need a condition of stationary data to be applied but not on non stationary data, can a non stationary model be applied to stationary data?
Thanks

Taken that the issue is with the model of non stationary ARCH without intercept where it is use for non stationary data. As the model is derived by setting the intercept = 0, 
so does it means that the same can be applied to the stationary model where we set the intercept = 0 to be applied to the stationary data?
[An ARCH model without intercept]
Christian M. Hafner, Arie Preminger
"In the stationary case, a well-known feature of classical ARCH
models is that the intercept is notoriously difficult to estimate, as
it is often very close to zero, and numerical difficulties arise for
the inference. This problem is more dramatic in more complex
ARCH-type models, where alternative estimation strategies have
been proposed such as the variance targeting idea of Engle and
Mezrich (1996). In the non-stationary framework of Jensen and
Rahbek (2004), a direct analog of this idea is not available, since
the unconditional variance of the process does not exist."
 A: 
Non stationary model applied to stationary data


If the model is non stationary, can it be applied to stationary
data/series?

Consider that "non stationary model" is an unfortunate definition. What can be stationary (in some sense see here: Stationarity and Ergodicity - links) or not is the process followed from your data. I make this clarification because I fear that It is not a semantical error only.

For example the KPSS test shows that the data is stationary but the BP
test shows presence of heteroscedasticity.

consider that KPSS test do not deal with homo/heteroscedasticity.

Taken that the issue is with the model of non stationary ARCH without
intercept where it is use for non stationary data. As the model is
derived by setting the intercept = 0

the kind of non-stationarity you have in mind have few to share with test like ADF or KPSS. The kind of non-stationarity considered in the phrase you cited is about series that are stationary in mean but not in variance. In this case any $ARCH$ specification is not adequate. In any case forcing the $ARCH$ constant to $0$ make no sense, in fact in this case the unconditional variance would be $0$. Indeed model like $IGARCH$ can help (read here for an example: https://www.oreilly.com/library/view/analysis-of-financial/9781118017098/c03_level1_6.xhtml).
A: You can refer to the link Forecasting Time Series: Stationary vs Non-Stationary.
For your question, I believe that it may can be used for stationary data but with a several impact. As what been discussed in the previous question in the link, certain model are not limited to specific type of data but it might come with a price. The ARIMA (0,1,0) for example can be used for non stationary data even though most are familiar with using it for stationary data. 
Hence as for the implication, the outcome might be different as what been described for long run and short run. By using a model that is not specified to be used might cause the model outcome to be problematic in the long run but for the short term, the outcome maybe can still be used. 
A: You might need to explore the stationary and non stationary definition. By definition,  a stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time while non-stationary process has a variable variance and a mean that does not remain near, or returns to a long-run mean over time.
As your question is regarding the ARCH model, maybe you could refer to https://freakonometrics.hypotheses.org/13547 and the article Stationarity, Mixing, Distributional Properties and Moments of GARCH(p,q)–Processes
(Alexander M. Lindne). There you could see more in detail the process of the model build. The article stated that if the sign of the Lyapunov exponent is negative, it implies stationary while non stationary if the sign change. This is a more clear definition I believe as the ARCH model are mostly dealt with the Lyapunov exponent.
Therefore, you could try to determine the stationarity first using the Lyapunov exponent and as I stated before, it is possible for the model to be used, but it might come with a price as stated by whuber in previous comment. As for validity, more research might need to be conducted to verify it.
