I thougt that Siddarta Gautama lived in a Hindu society. But were there Brahmans who disagreed with Siddartha or was Siddartha ever in an argue with them?
I thougt that Siddarta Gautama lived in a Hindu society
As far as you mean he lived in India, immersed in India culture of the time, yes. But as far as
hindu refers to an India majoritarily submsersed in the Vedic tradition, it's not clear this is so.
The period of the Buddha and early Buddhism is marked by a variety of religious movements known as sramanas (which included Buddhists and Jains, for example, among many others). So there was already a culture that rejected the vedic traditions of the time. Also, the geography seems to be of a region with many individual states and kingdoms, not of a "single India". I'm not sure if these states were, largely, supporters of brahmins or not (and if they were, it was soon in decline with the rise of the empires -- from the Nanda empire which seems to have supported the Jains, through the Maurian empire, when the emperor Ashoka converted and supported Buddhism).
But were there Brahmans who disagreed with Siddartha or was Siddartha ever in an argue with them?
Buddha disagreed not only with Brahmins, but with all the other sramanas whose teachings were not in agreement with the Buddha's teachings. These encounters, philosophical disputes and criticisms are massively documented in the buddhist canon.
Has Buddha Gautama ever had problems with some authorities about his teachings and way of live?
I don't remember the Buddha ever been in legal trouble, though sometimes kings came to him to solve some issue involving the sangha.