What are the advantages and disadvantages of having a branch of government that is not democratically elected? would you change the way the court is appointed?

Advantage is that they don't feel obligated to do what they think will be popular with the people, they will do things that are for the benefit of the country even if it's controversial. The disadvantages are how do you select the members, and if they do something unpopular, will they be recognized as legitimate as they aren't elected officials. I think you need to answer that last question as it is asking for your opinion