Cointegration if both variables are I(0)? I think I understood following so far about testing a cointegration relationship in a time series:
Two time series of same order of integration:
x: I(1), y: I(1)

Apply: Johansen and Juselius (1990) cointegration test
Two time series of arbitrary order of integration:
x: I(0), y: I(1)

Apply: Pesaran et al. (2001) ARDL bounds cointegration test.
Cointegration exists if a linear combination of two time series is stationary.
Question:
Can I theoretically apply the cointegration test to two stationary variables (trend-stationary, at level)?
x: I(0), y: I(0)

I would assume that a linear combination of two I(0) time series must be stationary, too.
Lütkepohl and Krätzig (2004) state in their book "Applied Time Series Econometrics": 

"Occasionally, it is convenient to consider [cointegration] systems with both I(1) and I(0) variables. Thereby the concept of cointegration is extended by calling any linear combination that is I(0) a cointegration relation, although this terminology is not in the spirit of the original definition because it can happen that a linear combination of I(0) variables is called a cointegration relation."

 A: Since

calling any linear combination that is I(0) a cointegration relation <...> is not in the spirit of the original definition

I will stick to defining a cointegrating combination as one where none of the original variables are I(0).
The case of two series:


*

*Two series of different orders of integration will never be cointegrated.  

*Two series both being I(0) cannot be cointegrated.  

*Two series both being I($d$) for $d\geq 1$ can be cointegrated, but they don't have to.


The case of more than two series:


*

*Only I($d$) series with $d\geq 1$ can enter the cointegration relationship. Thus I(0) series will never belong in a cointegration relationship. 

*At least two of the series must share the highest order of integration to cointegrate. E.g. three {I(2), I(2), I(1)} processes can cointegrate but {I(2), I(1), I(1)} cannot.


All of this is basic material that should be found in most of the time series textbooks dealing with cointegration. But I understand that keeping track of the discussion in a textbook can be hard, so let this serve as a summary.
A: There is no point in searching for a cointegration relationship between two stationary variables, I(0).
When the time series are integrated of order one (or higher), standard linear regression analysis may produce spurious results; in particular, we may erroneously find that one variable is statistically significant for explaining the other variable (Granger and Newbold 1974) [1]. This is where cointegration comes into play. 
The focus of cointegration analysis is to search for a linear combination of the time series that has a lower order of integration than the original series. For example, in economic data, it is sometimes observed that the variables are I(1) (stochastic trends) but a linear combination of them is I(0) (stationary). The cointegration relationship reveals a stable relationship between the variables in the long-term and this becomes helpful to develop further economic analysis or to test economic theories.
If the variables are stationary, I(0), then we can rely on standard linear regression analysis and test, for example, whether one variable is a significant predictor of the other variable. (Some kind of cointegration relationship between I(0) variables would not be informative, since any combination of stationary variables will, in principle, be stationary.)
[1] Granger, C. W. J. and Newbold, P. (1974). "Spurious Regressions in Econometrics." Journal of Econometrics, 2(2), pp. 111-120, DOI: 10.1016/0304-4076(74)90034-7.
