In principle, I would say that there is a clear room for the application of PCA, whose aim is just to reduce the dimensionality of the dataset. Perform the PCA and keep the first component. Try to find also some interpretation for this first componente. Then just plug in your econometric specification.

Check our article, it uses the PCA to obtain a low-rank approximation of the weakly stationary, multivariate time series processes.

Article Block circulant matrices and the spectra of multivariate sta...

Yes you can use PCA to break the system of variables to a smaller set. You obtain the eigenvalues of either the covariance matrix or the correlation matrix. Arrange the eigenvalues in descending order of magnitude. The first principal component is associated with the first eigenvalue. The second component with the second, and so on.

Keeping the first component is surely not enough. An alternative to PCA is dynamic factor models which exploit the time series nature of the variables.

did you already consider learning a VAR model (parameter matrices) and reading out causality form the zero-pattern of the parameter matrices:

https://nowak.ece.wisc.edu/smart_arxiv.pdf

Yes, you can use PCA.

Regards

Yes, you can use PCA on a large VAR. There are lots of articles on Google Scholar about this.

Yes you can. You can use lasso also.