Why is it assumed that $x[n]$ is limited from $0$ to $N-1$ while evaluating DFT?
How to evaluate DFT for a sequence extending from n=2 to say n=N+1?


Formularistic: trivial variable substitution of the running variable $n$ by $\tilde n = n+2$, done. 

Looking at it as a signal: When doing an DFT, it helps to think of it simply as mapping of $\mathbb C^N\mapsto\mathbb C^N$, i.e. a complex vector goes in, same size complex vector comes out. 

Your "shifted" input hence is simply a different input value for the DFT.

Now, when we interpret the DFT as a mapping between discrete time- and discrete frequency-domain, we have to assume that the time signal is periodic with the DFT length. That means the two "new" values are just the oldest two values. I.e., your reindexing just leads to a circular shift of the values in the vector.