https://terrytao.wordpress.com/2011/06/25/two-small-facts-about-lie-groups/
In this blog post, Terence Tao mentions that 'If $G$ is a compact connected Lie group, then the exponential map is surjective'. It seems to me from this statement that every element of a compact connected Lie group has some corresponding element in the Lie algebra $\mathfrak{g}$, though the exp map may be many to one. However I have also seen the following statements
1) 'A simply connected Lie group is determined by its Lie algebra...'
2) 'If $G$ is a compact, simply connected and connected, then arbitrary $g\in G$ can be written as $g=e^X$, for some $X\in \mathfrak{g}$.'
These seem to contradict Terence Tao's statement, since now it seems we need the extra condition of being simply connected for elements of $G$ to be determined by $\mathfrak{g}$. What is the reason for this contradiction? (Perhaps when $G$ is simply connected, the map is locally injective?)
You need simply connectedness for uniqueness. For example, both $SU(2)$ and $PSU(2)$ are compact connected group with lie algebra $\mathfrak{su}(2)$. One is covering of the other.
On the other hand, your statements 1) and 2) are called Lie's third fundamental theorem: For a lie algebra $\mathfrak{g}$ of finite dimension over $\mathbb{R}$ or $\mathbb{C}$, there is a connected simply connected lie group $G$ such that $\mathfrak{g}=lie(G)$, unique up to isomorphism.
The proof is by embedding the finite dimensional lie algebra into some $\mathfrak{gl}(n,\mathbb{R})$ and $exp(\mathfrak{g})$ can be seen as a subgroup of $GL(n,\mathbb{R})$. The uniqueness part involves Lie's secord fundamental theorem which uses simply connectedness.