Translation:She is in Russia, but is already going home.
идут has a sense of motion by walking, while едет has a sense of motion by car or other means of transportation. In this sentence is said that she is in Russia and she is already going home, so probably her home is outside of Russia and she couldn't go by walking, but need some kind of transportation. I hope this clarifies your doubt! :)
Yes, that's the difference. If you walk or at least visit your usual places within town ("work", "school" etc.) it is я иду. If you travel using some form of transportation, it is я еду.
The infinitive is ехать. All non-past forms are formed with ед- (еду, едешь, едет, едем, едете, едут).