1. Mr X was the first president of independent Poland. Can I put "the" before "independent" i.e. "...of the independent Poland" If not, why? Yes, you can. I would do so in a context where I wanted the reader to bear in mind that there was also, perhaps at another time in history, a Poland that was not independent.
1. I would leave it as is. He was the first president of Poland. I don't sense independent as restrictive, but as non-restrictive -- president of Poland, which happened to be independent at the time. I can't imagine what the first president of the dependent Poland would have been. How could he have even been a real president? It almost suggests that both Polands could have
It often happens that even native speakers, who often tend to rely rather on intuition than on rules, differ in their opinions as whether to use an article in a given situation or not.