What is Itachi's illness? What was Itachi's mysterious illness? 
Do you think Tsunade could have cured it? I'm aware that it is a terminal illness.
And when did Itachi get this illness?
The disease is never named.

This is actually quite common in manga/anime. It's often said that a character has some disease without the name of the disease ever being stated.

In some cases, the disease may be implied, such as in the case of said character suddenly being bald, in which case cancer is being implied. This can give a story some realism

In other cases, no symptoms are given or the symptoms are very generic, so that it's impossible to determine what disease the character has. This is done because it means less work on the author's part if they're not familiar with a disease that has the symptoms they want and means less time has to be spent developing things that don't have any relevance to the story.

In this case, very little information is given about Itachi's illness other than that it's terminal. It's likely that the author did this simply because what Itachi had didn't really matter to the story in any way. The only important thing is that it forced Itachi to speed up his plan, knowing he would die.

There is one other thing we know about his disease. According to Narutopedia, this disease had an effect on Itach's chakra. Because of this, it is reasonable to assume that this disease either isn't real or it's based on a real disease with this fictional characteristic attached.