Interpretation of regression coefficients with and without logs
A possible answer if it's not too late.

Q1: Yes, if you mean "per years of schooling" at the end of your question. So, the unit of $\beta x_i$ is $\$/hr$.

Q2: The unit of $\beta$ is $\log(\$/hr)/years$ literally. Then the unit of $\Delta (\beta x_i)$ is $\Delta \log(\$/hr)$, nothing strange, which is approximately the % change times 0.01. This is "dimensionless" if you like to say so, or would "dimension-irrelevant" be better? We interpret $\Delta (\beta x)$ as % change, not $\beta$.