A dominate morphism is not an isomorphism Here is a Theorem: Let X is a smooth variety of dimension 1, and let f:P$^1$$\to$ X be a dominate morphism. Then X is an isomorphism to P$^1$(although f need not be an isomorphism). I want to find an example that the above-mentioned f is a dominate morphism, but not an isomorphism.

In characteristic zero, this is a consequence of Hurwitz's theorem $$2g_ {P^1}-2=n(2g_ X-2)+deg( Ram(f))$$ which in your case yields $$-2=n (2g_ X-2)+ \text {nonnegative number }$$
This equality forces $g_X$ =0 (else the right-hand side is nonnegative) and since the only smooth projective curve of genus zero is (up to isomorphism) $\mathbb P^1$, your first question is answered.

As to your second question, just take $X=\mathbb P^1$ and $f([z:w])=[z^2:w^2]$