Does chemotherapy cause other health problems?

Possible. Some chemotherapy such as taxanes and vincristine can cause neuropathy (nerve damage) others such as the vinca alkaloids can cause constipation. Gemzar (gemcitabine) can cause diarrhea. Cisplatin can cause hearing loss and neuropathy and kidney damage. Some of these effects are short and some long term complications. There are many other drugs which are used. Speak with you oncologist regarding more specifics.
Yes. All chemo drugs have side effects, some short term, some long term. Overall, the benefit significantly outweighs the risk, but some long term problems include the potential for heart damage, nerve damage or an increased risk of a second cancer. The risks are different with each of the drugs and should be explained by your doctor before starting treatment.