2) This is, of course, the same as saying about spacings between uniform points on a segment (you can say that $Y_1=0$, for example). Let it be the segment $[0,1]$.

Now the joint distribution of $I_1,\dots, I_{n}$ is the same as of $E_1/E,\dots, E_n/E$, where $E_1,\dots, E_n$ are iid exponential distributed, $E=\sum_{k=1}^n E_k$ (see Devroye *Non-Uniform Random Variate Generation*, p.208). So the distribution of $I_{(1)},\dots, I_{(n)}$ is the same as of $E_{(1)}/E,\dots, E_{(n)}/E$. But the joint distribution of $\{E_{(k)}-E_{(k-1)},k=1,\dots,n\}$ ($E_{(0)}:=0$) is the same as of $\{(n-k+1)^{-1} E_k,k=1,\dots,n\}$ (ibid, p.211). 

So the distribution of $J_1,\dots, J_n$ is the same as of $\{(n+k-1)^{-1} E_k/E,k=1,\dots,n\}$, where $E_1,\dots, E_k$ are iid exponential rv's, $E=\sum_{k=1}^n E_k$. And this is, by the previous paragraph, equivalent to saying that the distribution is the same as of $\{(n-k+1)^{-1} I_k,k=1,\dots,n\}$.

These are not independent, but very close to be, and from here you can find the distribution of maximum and minimum (but nothing very pleasant there, as the variables in question are not identically distributed; a formula for the expectation looks extremely ugly).