Does the quantile mapping procedure ensure that by changing e.g. solar radiation values the other parameters influenced by solar radiation (e.g. temperature) are corrected accordingly?

It is possible to use a bivariate approach (eg DOI: 10.1002/env.2454) but these methodologies are rather complex and don’t always work as expected. That’s why we used a univariate approach.

I understand that you had to correct both parameters independently, right? Are the dependencies still correct then after correction?

Unfortunately the cross dependencies are not assured. A bivariate approach (multivariate would be even more complex as you probably know) would be preferred if this is a critical thing for you.