On the Traefik dashboard it looks like everything is correct configured, but when I try to call app.server.com/api/ the nginx behind "/" gets the request and throws a 404.
No, as far as I understood it, it's not necessary, because the "/" rule already creates the certificate for the domain. You think, that may be the issue?
It probably will be useful to familiarise yourself with some core concepts. Do not just read the overview, read also all the subsections. As cakiwi noted, there is not concept of "subfolder" in traefik, there are routers and services, and naturally you do need a separate router in your use case.