1) Your interpretation of $\beta$ as a semi-elasticity is correct. 

2) You know $$\beta \approx \frac{\frac{Y_t-Y_{t-1}}{Y_{t-1}}}{\frac{X_t-X_{t-1}}{T_{t-1}}}$$ 
An elasticity is
 $$\epsilon \approx \frac{\frac{Y_t-Y_{t-1}}{Y_{t-1}}}{\frac{X_t-X_{t-1}}{X_{t-1}}} = \frac{\frac{Y_t-Y_{t-1}}{Y_{t-1}}}{\frac{X_t-X_{t-1}}{T_{t-1}} \cdot \frac{T_{t-1}}{X_{t-1}}}=\beta \cdot \frac{X_{t-1}}{T_{t-1}}.$$

This is a function that depends on the ratio $R_{t-1}$ of $X_{t-1}$ to $T_{t-1}$ for each $i$. You can average that over your panel to get an average elasticity, which simplifies to $$\epsilon = \beta \cdot \bar R.$$

Since you are effectively conditioning on $R_{t-1}$ in your model, this is just a random variable multiplied by a non-stochastic constant, which makes the variance of the elasticity easy to calculate:
$$\mathbf{Var}(\epsilon) = \bar R^2 \cdot \mathbf{Var}(\beta)$$