Yes, stationarity is essential feature in time series framework. But, when you go with the panel framework you start with several other issues. Botom line, you shoul see what is with the cointegration. You may have two nonstationary time series, but linear combination of them could be stationary.

You have to have on mind ergodicity. Stationarity is something that you need in context of time series on your path to ergodicity. And issue with panel is that you may have relation between units in a cross section. But, sometimes that means that "two non stationarity time series jointly give stationarity process" (intuitively speaking). But, further, you may have problem in context how the data are affected by each other a cross sections. You need have much more potential issues in panel data than in classical time series approach. If your potential problem was one dimensional, now it has two dimensions.

yes it is essential to avoid pseudo regression.

time series and panel modeling contain some steps. first step is stationary checking.

stationary must be conducted before any regression with time features, Regardless of model name . if data is not stationary, you should try co-integration. if your data sets are not co-integrated and theory of your work said that there is relationship between variables, you can shift to models in which stationary is not important.

bests...

No. If you have time series with a unit root, you can use test for stationarity and cointegration with time series I(1). Afte you can use FMOLS, DOLS or CCR for analyze the cointegrarion and causality. You can see Baltagi: "Econometric analysis of Panel Data". Chapter 12 o my papers about Wagner´s law using panel data (it is disposable on my page in ResearchGate).

Regarding cointegration test. There are three main types:

1. residual-based tests for the null of cointegration

2. residual-based tests for the null of noncointegration

3. tests based on vector autoregression

In Oxford Handbook of Panel Data you also have the paper related to panel coointegration by In Choi.

we need a structural process in the data before we start testing short term causality even if in panel date

Assuming two series of interest: A non-stationary series cannot be Granger-causal for a stationary series. Perhaps a stationary series could be causal in the broad sense for a non-stationary series, maybe even Granger causal if you extend the details of Granger's framework but the standard tests do not accommodate this. Conclusion: the two series should have the same level of integration. Hence, if one is stationary then you either difference the other or integrate the stationary one.

Most tests assume stationarity, even in panels. My belief is that the wider the panel gets then the less is the risk of spurious regressions with non-stationary series but, even then, the distributional theory that underpins standard tests may need revision.

So, if you have only two series, best make them stationary, and satisfy yourself that causality makes economic sense in that context. If more than two series then the possibility of cointegration means that all of the above needs revisiting.

No its not necessary u have just to follow method unit root tests cointegration tests and fmols dols imols or CCR and causality tests if u want. So u don't have to manage data to be stationary. We made that only for pvecm.

Yes, stationarity is a necessary condition in time series analysis. Particularly if you are planning on estimating coefficients using panel cointegration tests.

Yes, data must be stationary. May be this paper will help

https://www.unine.ch/files/live/sites/irene/files/shared/documents/Publications/Working%20papers/2017/WP17-03.pdf

Stationarity of variables is a necessary condition in time series analysis. Also, if you are planning on estimating coefficients using panel cointegration tests.