Not quite.
Your interpretation for the first model is correct, but your explanation isn't quite right. Note that
$$ \begin{equation*} \beta_1 = \frac{\partial \log y}{\partial x}, \end{equation*}$$
but that isn't very easy to interpret. So, we recall the calculus result that
$$ \begin{equation*} \frac{\partial \log y}{\partial y} = \frac{1}{y} \end{equation*}$$
or
$$ \begin{equation*} \partial \log y = \frac{\partial y}{y}. \end{equation*}$$
Plugging this into the equation for $\beta_1$, we have
$$ \begin{equation*} \beta_1 = \frac{\partial y / y}{\partial x}. \end{equation*}$$
If we multiply both sides by 100, we have
$$ \begin{equation*} 100\beta_1 = \frac{100 \times \partial y / y}{\partial x}. \end{equation*}$$
We realize that $100 \times \partial y/y$ is just the percentage change in $y$, giving the interpretation that $100 \beta_1$ is the percentage change in the outcome for a one unit increase in $x$.
The correct interpretation for your second model would be that a 1 unit increase in GDP leads to a 10 percentage point increase in sales. It's easiest to understand this by thinking of your outcome as being measured in percentage points, rather than percent. Then, a 1 unit change in $x$ leads to a $\beta_1$ unit change in $y$, just as we normally get.
This is an important distinction. An increase in sales from 1% to 5% is a $5 - 1 = 4$ percentage point increase, but a $(5 - 1)/1 \times 100 =400$ percent change.
1) Since it is an odds ratio it doesn't matter where you start. The odds for an 18 year old are 3 times those for a 17 year old. Or the odds for a 17 year old are 1/3 those of an 18 year old. Same thing. If you want to get the probability that a person of a particular age will be employed, you can use the formula with the parameter estimates (not the ORs). Or you can get the program you are using to do it for you.
2) Whether centering helps is a matter of opinion. I don't find centered models clearer, but some people do.
3) The odds are not exactly the same as "likely" (although many people speak as if they were) and the odds for a 17 year old would be 27 times those of a 14 year old.
Finally, I'd be cautious about this model. The model assumes that the OR is the same between 14 and 15, 15 and 16 and so on. That seems unlikely to me, based on what I know about the subject.
Best Answer
Pretty much. But:
Holding $height$ constant as long as $height\not=0$. If it equals $0$ than the interaction term is also $0$. Also, if $height>0$ and held constant, than for every increased unit of $age$ , the average predicted score will increase by $(0.4+0.8)$, not by $0.4+0.8*height$.
$\beta_1$ (0.4) will be used alone only when $height=0$, or else the interaction will be also coming into play.
The same principle goes, naturally, also for both covariates.