I will try to present some quick facts about *exponential polynomials* and try to address your questions, however you really ought to look at the literature, particularly starting from Ritt's work in the following articles

[1] J. F. Ritt, "A factorisation theory for functions $\sum_{i=1}^n a_i e^{\alpha_i z}$", Trans. Amer. Math. Soc., 29, 584–596, 1927

[2] J. F. Ritt, "Algebraic combinations of exponentials", Trans. Amer. Math. Soc., 31, 654–679,
1929.

[3] J. F. Ritt, "On the zeros of exponential polynomials", Trans. Amer. Math. Soc., 31, 680–686,
1929

First a few words about factorization of such functions. I will be assuming the underlying field for the coefficients and exponents is $\mathbb C$, though this works over any algebraically closed field. An equivalent representation of these "fractional polynomials" is in the form $\sum_{i} a_i e^{\alpha_i z}$. First note that we have all these units of the form $ae^{\alpha z}$, and moreover, as you noticed, binomials $1-ae^{\alpha z}$ are divisible by $1-a^{1/k}e^{\alpha z/k}$ for all $k\in \mathbb N$, so to have unique factorization one must somehow get around these examples. This was done by Ritt [1] in the case of constant coefficients $a_i$ and by Everst and van der Poorten in "Factorisation in the ring of exponential polynomials", for the general case when $a_i$ are polynomials in $z$.

Let $\mathcal W$ be a finitely generated $\mathbb Z$ submodule of $\mathbb C$, and let $\mathbb C_z\lbrace\mathcal W\rbrace$ denote the ring of exponential polynomials of the form $$E(z)=\sum_{i=0}^m a_i(z)e^{\alpha_i z}$$
where the $\alpha_i$ are distinct. Now one can show that unique factorization holds in this ring.

**Theorem:** An exponential polynomial $E\in \mathbb C _z\lbrace\mathcal W\rbrace$ factors uniquely up to units as a product of a polynomial $A_0(z)$, a finite number of polynomials $A_i(e^{\beta_i z})$ where the ratio of any two $\beta_i$'s is not rational, and a finite number of exponential polynomials that are irreducible in $\mathbb C _z\lbrace\mathcal W\rbrace$.

In [2] there is a proof that if all zeroes of an exponential polynomial $E$ are also zeroes of the exponential polynomial $F$ then there is an exponential polynomial $G$ and a polynomial $A$ so that $E(z)G(z)=A(z)F(z)$.

Now you should look at [3] for information on zeros of exponential polynomials. Here are two particularly nice facts. The first is an analog of the fact that a polynomial of degree $n$ can have up to $n$ roots. Let $P$ be the smallest convex polygon containing all the frequencies $\alpha_i$ of an exponential polynomial $E$, defined as above. Let the sides of $P$ be denoted as $b_1,b_2,\dots,b_k$.

**Theorem:** (Tamarkin, Polya and Schwengler) There exist $k$ half strips, with half rays parallel to an outer normal to $b_i$ which together contain all zeroes of $E$. If $|b_i|$ denotes the length of $b_i$, then the number of zeroes in the $i$'th half strip with modulus $\le r$ is asymptotically equal to $\frac{r|b_i|}{2\pi}$.

The second fact (from [3]) is a statement which is an analog to Viete's formula for the case that the product of the roots of a monic polynomial is the constant coefficient, up to sign. Ritt proves that for an exponential polynomial of the form
$$f(z)=1+a_1e^{\alpha_1 z}+\cdots +a_ke^{\alpha_k z}$$
with $0 < \alpha_1 < \cdots < \alpha_k$ are real. Let $R(u,v)$ be the sum of the real parts of those roots whose imaginary part lies in $(u,v)$. Then one has
$$R(u,v)=-\frac{(v-u)\log|a_k|}{2\pi}+O(1).$$