The answer is almost certainly a solid: No.
The length of time that a client is willing to wait until it receives a response from the server is fully configurable by the client. Either the HTTP library being used by this app has a short time set by default, or the app itself has configured the request to have a short timeout. For an example, see this ...
That is not wrong, but redundant. As you say, if there is a legacy
system involved, then such an approach might be unavoidable.
Effectively what you have an initial authentication, which is later
superseded by authentication from sub-service.
Yes, X-Auth-Token is
an appropriate header for your secondary token.
The only point to consider in this case - what ...