There are a lot of stuff available on internet related to service and it's usage, But still, I will try to answer as much as possible.
Service: It is android component where it runs on background and foreground of the system.
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.
Android - implementing startForeground for a service?
A background service performs an operation that is not noticeable to
the user, It can be downloading any file. We can use this for long
running operation.
Coming back to your question, will try to answer one by one.
Will the service resident in mobile phone system until I uninstall the App ?
Yes, It will reside on your phone until you not uninstall application. But it will not run infinite, As your job finished it will be stop.
Will the service cost may system resource when it stay in system?
It may cause, If you are not using it properly, If you are running background service we should know when to stop it, unless it will run continuously. It may also cause you memory related issue.