Using below graph API you can get the chat between user and teamsApp.
GET /users/{user-id | user-principal-name}/teamwork/installedApps/{app-installation-id}/chat
Ref Doc: https://learn.microsoft.com/en-us/graph/api/userscopeteamsappinstallation-get-chat?view=graph-rest-1.0&tabs=http
For sending mail you can use the sendmail API.
POST /me/sendMail
POST /users/{id | userPrincipalName}/sendMail
Ref Doc: https://learn.microsoft.com/en-us/graph/api/user-sendmail?view=graph-rest-1.0&tabs=http
Thanks,
Nivedipa
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