I am trying to convert an Odds Ratio to an Effect Size (ES). I found this article from Cross validated that provides a way of converting a Log Odds Ratio to Cohen's d but am unsure of three things.

1. Is an 'Odds Ratio' is the same as a 'Log Odds Ratio'?
2. Does the statistical significance test and heterogeneity tests directly translate to the converted ES or do these need to also be converted?
3. I am also unsure if I am carrying out the calculations correctly. Based on three studies the OR is 0.52 (CI = 0.37, 0.74; p = 0.0002) Heterogeneity: Chi2 = 3.66; I2 = 45%. Does my calculated/converted ES below seem accurate? I don't mean to ask someone to do them for me so have tried to figure them out myself but I'm not 100% sure if I've done it right.

Cohen's d = 0.287 (CI = 0.204, 0.407)

Thank you for any guidance.