Can asthma go away and then come back when a child leaves and returns to the city?

Yes. While asthma probably does not go away, symptoms may very well come and go depending on the presence of triggers. If your trigger is air pollution, for example, you may see asthma "return" when moving back to the city.
Yes. Asthma is an inflammatory lung disease that reacts to allergens, activity, irritants, and other things. It is not uncommon for one location to exacerbate asthma symptoms, while others do not. Smoking, car exhaust, higher ozone levels could easily cause trouble, while getting out into the country might ease it. Assuming grass and tree pollen isn't a problem then yes.