When I want to use Google's Gmail API within my web application in order to receive and send emails, then must this web application be deployed in the Google Cloud as a precondition and any on-premise hosting will fail? Is this the price one must pay to use it?
Your application's code can be hosted anywhere you want. However, you do need to create a Google Cloud account to create a project, enable the APIs and get the application credentials:
Cloud APIs use application credentials for identifying the calling applications. Credential types include API keys, OAuth 2.0 clients, and service accounts. You can use Google Cloud console to create, retrieve, and manage your application credentials. For more information about application credentials, see Authentication Overview.
Once you have your project's credentials you can just create the code within your current app and use the credentials wherever they are needed. You can refer to one of Google's quickstarts for that.