Mahabharta- Who was the rightful heir of the Kingdom. Udhistra or Duryodhana?Dhritirashtra was the legal heir of the kingdom but Pandu was made...

This is a very interesting question with no clear cut answer. Explanations to your question already point to the reason for this confusion. Dhritarashtra and Pandu were both sons of King Vichitravirya. As per the traditions of that time the elder son of a king normally ascends the throne after the father steps down or dies. In case of Dhritarashtra, in spite of being the elder brother, he was not made the king because he was blind. It was thought that being blind, he will not be able to manage the duties of the king effectively. Therefore Pandu, the younger brother was made the king. However Pandu died at a young age, and at that time it was decided to make Dhritarashtra the king, in spite of his blindness.

So far there is no confusion. But when it comes to deciding between Duryodhana and Udhishtir, both have some claim over the throne, both being eldest sons of Kings. I would personally say that Duryodhana has greater claim to throne on two counts. First, his father was the original rightful heir to the throne. Second he was the ruling king, who has precedence over all past kings.

But don't let this discussion confuse you over the cause of the war between Pandavas led by Udhishtir, and Kauravas led by Duryodhana. The fight was not over the issue of inheriting the kingdom or the throne. Pandu never asked Duryodhana for the throne of the entire kingdom. He only asked for reasonable control over part of the land within the kingdom, to which he and his four brothers were entitled as princes.  Duryodhana refused to accept this reasonable demand, saying that he will not give land even as small as point of a needle. In such circumstances Udhishtir, as a Kshatriya, was within his rights to win the whole kingdom in a war. Which he did and thus became entitled to the crown of the whole kingdom.

Regarding the use of the name Kaurava, it is quite right that both Kauravas and Pandavas were descendents of King Kuru, and bore the same relationship with him, but for some reason not known to me, only Kauravas are associated with that name. One possibility is that Pandavas won the Kingdom of Kuru Dynasty in war, rather than claimed it as an dynastic inheritance.