“on the right foot” is not a phrase you can adapt to other situations; you really should use it with “get off on the right foot” to mean “to begin well.” But you can say, “Some countries got off on the right foot, unlike some others.” That would mean, some countries started well. However, with countries … most places have histories going way back before we even had a system of countries and nations. You might be able to specify a specific situation that began well for some places. For example, “Some countries got off on the right foot distributing vaccines, while others did not.” That would mean, some countries began distributing vaccines well, while others did not.

Great idea. This might sound a little more natural, but you’ve got the right idea: “Even in professional settings, the best way to get off on the right foot when introducing yourself is to give a few personal details about your own life.”

Great example. I didn’t mention this in the lesson, but you can say, “get the [time period] off on the right foot.” So for example, get the day off on the right foot, get the year off on the right foot. Therefore, you can say, “Usually people want to get the year off on the right foot and promise to do new things or stop doing other things.”