One thing to always be mindful of, know who the intended audience of the writing is aimed at, because it will help you in understanding it better.
The Didache was written to established Christian communities who had already been visited by Apostles and other disciples and those who had built them up. Naturally they would need to stay for an extended time to instruct them in the Christian faith. This document is teaching them after this event. The concern is with people who would mooch off a community claiming to be an Apostle. A true Apostle would not need to stay in an established community for very long. They were always looking to expand the message of Christ, so they would rest and recupperate with them, and then move on to evangelize new places/peoples.
Those sections are warnings on how to spot false teachers/prophets/Apostles so they wouldn't be taken advantage of, especially since all Christianity was being persecuted, and it could be difficult to truly know who was who since they worked very hard to maintain necessary secrecy at times.
As to supporting prophets (chapter 13), nothing in there says that Apostles or prophets can't support themselves, only that the community should support them as they generally are so busy spreading the Gospel that they are unable to earn a living. They are distinctly noted separately from regular Christians who were supposed to earn their keep (chapter 12).