"Why is Kamari the only offered to be a High Protector? Can other condemned be offered a similar position?"

The Elders would reason it's because the fates chose Kamari to prove faerself free of the fallen fairy traits that plague most of faer kind. If Kamari protects the others from fairies like faerself, dangerous fairies, then fae's offset the danger of faer existence. To Kamari... it implies fae's inherently a dangerous fairy and fae spends a lot of time trying to come to terms with that and eventually assumes its something everyone else thinks of faer (the play) but it's not true. Fae's only using the High Protector to prove that.  

As for the actual reason why the Elders chose Kamari to by the High Protector... they wanted absolute desperation. Kamari has nowhere else to go to, no one else like faer, and even the most discriminated against fairies still see Kamari as beneath them, and they wouldn't dirty one of their own claws with fallen fairy blood. So better to take the worst of the worst, practically tree-less, and make faer into the executioner. The title won't give faer enough power to change the way others see faer, it won't give faer the power to do anything other than the Elder's orders... and that's the point. It's something more than what fae had before. Some Elders tell themselves it's a kindness to Kamari... 'we're give you all this and can't you give us this small thing in return?' small thing includes having to wretch younglings away from their guardians