Smallest field extension of $ℚ$ that is closed under $\exp$

Question

Let $F$ be a field such that $ℚ \leq F \leq ℝ$. Let $G = F(\pi,i)$. I want to define $\exp$ function on $G$. It must satisfy that:

1. $\forall x,y \in G \space \exp (x + y) = \exp x \times \exp y$

2. $\exp 0 = 1$

3. Periodicity and Positiveness: $\exp \pi i = -1$ and $\neg \exists x \in G \cap [0, \pi) \space \exp xi = -i$

4. Existence of Logarithm: $\forall x \in G^\times \space \exists y \in G \space \exp y = x$

My questions are:

1. Is $\exp$ uniquely defined by above?

2. What is the smallest such $F$?

3. Is the smallest such $F$ computably enumerable?

4. Given an oracle that computes $\exp$, is $\log$ a computable function (if not nessesarily assuming a branch cut)?

5. What is an example of a computable number that is not in $G$?