Whenever someone checks the code on TFS, I want to receive a notification. Is there any way to do this using the API?
In addition, I want to get information about the control, for example, files affected by / change types
Yes. This is how the default notification works by default. You can sign up for an event with bissubscribe, which can send an email or SOAP notification.
See here for more details, including more information on how to write a SOAP service that will handle the event.