Thanks for your query, I have gone through your query. The chances of root canal treated tooth getting infected is very less unless it is not properly treated. The fell out tooth structure could be only the crown portion. The Root canal treated tooth becomes weak after RCT, so it has to be crowned immediately. Now the roots might be remaining with in the bone. Consult an oral physician and get a radiograph to rule out the presence of root piece. If root is there then get it removed. I hope my answer will help you, take care.