Does K8s ClusterIP service supports custom timeout?
All cookies and headers are a construct of the HTTP protocol. Kubernetes Services operate on TCP/UDP packets therefore it will pass the payload as-is.

It is not that there isn't an nginx.conf file but rather the configuration is managed via annotations. See this link
https://github.com/kubernetes/ingress-nginx/blob/master/docs/user-guide/nginx-configuration/annotations.md