by natural in any smart phone you can have multiple apps running at once. When you have an app running, but it’s not the focus on the screen it is considered to be running in the background.
as for services there're 3 types of them (quoting from android developer documentations) :
Foreground
A foreground service performs some operation that is noticeable to the user. For example, an audio app would use a foreground service to play an audio track. Foreground services must display a Notification. Foreground services continue running even when the user isn't interacting with the app.
Background
A background service performs an operation that isn't directly noticed by the user. For example, if an app used a service to compact its storage, that would usually be a background service.
anyway, background services now have certain limits starting with android 8 (api 26) and it's recommended to use long-running-workers through WorkManager
Bound
A bound service offers a client-server interface that allows components to interact with the service, send requests, receive results, and even do so across processes with interprocess communication (IPC). A bound service runs only as long as another application component is bound to it. Multiple components can bind to the service at once, but when all of them unbind, the service is destroyed.
As for if your application having a service or not, if you didn't include/define one in the app manifest, then no your app doesn't have one by default as far as I know.
You can read more about services from this link
I don't know much about firebase, I just wanted to help and clarify some of your questions.