Can clindamycin cause heartburn and, if so, is this a serious side-effect?

All antibiotics. all antibiotics can cause heartburn,some more that others,specially if it was taking on empty stomach. it is not a serious side effect,not enough to stop the medicine. check with your physician.
Possibly. it can cause several GI side effects but they are not usually serious or worrisome.
Yes. Take the minimum dose for the minimum amount of time possible. Try taking it with food, sweet potato. Consume a probiotic for a month. Get sunshine (preferably) or take vitamin D3 10,000 IU/day.