Why is ww a linear indexed language while www is not?
In linear indexed languages, the intuition is that you cannot copy a big information in one step, you have to do it in aletter-by-letter way. This causes things like $ww^R$ much easier to do than $ww$. We can still do $ww$ because you can first "create" the word $w$ by pushing its letter onto the stack and writing them at the same time on the output. Then you can pop the letters to the right or to the left as you prefer, so you can always produce $ww$.
For $www$ it's not possible anymore, because after producing $w$ on the stack and writing in on the output, you need to duplicate the stack, which is not allowed. However you could produce $www^R$, by writing the letters on both sides when popping the stack.