This is easy. The DFT formula is
$X_k = \sum_{n=0}^{N-1} x_n \omega^{-nk}$ (forward transform) $x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \omega^{nk}$ (reverse transform)
where $\omega = e^{2\pi i/N}$ is a primitive N-th root of unity in the complex numbers. So just take $N = 4$ and you get $\omega = i$. Note that there are some variations in the above, namely with regards to the sign used: some authors may use $\omega = e^{-2\pi i/N}$.
Since the FFT just computes the DFT, we can just use this. For the first part of the question, to transform $x = (1, 0, 0, 0)$, we have
$X_k = 1 \omega^{-0k} + 0 \omega^{-1k} + 0 \omega^{-2k} + 0 \omega^{-3k} = \omega^{0} = 1.$
So the transformed sequence is $X = (1, 1, 1, 1)$. The other question can be answered by computing the inverse DFT of $X = (1, 0, 0, 0)$. The result is $X = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4})$.