Alumni Using Personal Gmail Accounts
When logging in to your Cornell G Suite account, you will use CUWebLogin (not Google Sign-In) with your NetID@cornell.edu address and password.
Currently, alumni users of Cornell G Suite are not required to use Two-Step Login, but are allowed to enroll to get additional security for their Cornell G Suite account. After enrolling in Two-Step Login, new alumni users should be sure to follow the steps at Expand Where You Use Two-Step Login.
The short answer is no. This is known as spoofing and it's a bad thing, even if intended harmlessly. Various email systems look for evidence of spoofing and may block messages and the accounts sending them. Neither Google nor IT@Cornell support this practice.
To summarize: if you want to send email that looks like a Cornell account, send it from your Cornell account.