Are there any asthma medications that don't upset the stomach as a side effect?

Yes. Most inhaled medications have no effect whatsoever on the stomach. Neither does oral singulair, (montelukast) for most people.
Depends. Very few asthma medications cause stomach upset in most patients; there are a wide array of preventive medications and rescue medications - work with your physician until you have a set that doesn't bother you.
Yes. Talk with your prescribing physician about the medicine you are taking, the side effects and the level of control of your asthma. In general most asthma meds are well tolerated but every patient is an individual and one med that has no side effects in one can not be tolerated by another.