1. Immunity, if it exists, will not last a lifetime. We can look at other coronaviruses in circulation and guess that there will be immunity, but that it won't last forever.
2. Reinfection will be less likely to cause a severe illness than the first time around.
3. Stories of reinfection are likely testing errors as you say. However, this doesn't mean that people won't get reinfected at some point. Likely they will. But we wouldn't expect them to be getting reinfected that quickly.
4. No one knows anything for sure. People may actually be getting reinfected quickly.