Zinc helps the system to cleanser and repair the lymphatic system. Lymphatic system expels waste and toxins from the body, so yes, it can stop body odor. Find out more from healthatoz.com.
No side effects unless more than 12 mg (daily allowance is exceeded for long periods of time. Most vitamins contain Zinc, but you can buy Zinc 50 mg tablets and take as directed
Too much zinc is toxic. You should know that. It's not a problem, unless you are doing something really stupid.
Zinc is often referred to as a scavenger. It cleans a lot of things inside your body (and other things as well) some of which cause oder. It will even work in ionic form outside your body, but not as well as multivalent metals (for some people these are a problem) . We are talking deodorants here.
At this point I bet you will be besieged with vitamin lovers to answer your question. Their information will be correct. But if you are eating a balanced diet, you are getting enough zinc. You have to work weal hard to avoid getting enough zinc. Taking more will not help.
Changing other things in your diet will change odor a lot more that taking zinc will. Take lesser meat,more vegetables and fruits,detox your body often to stay healthy!