What does connecting Google grant?
Connecting authorizes the assistant to read your Gmail (to triage and summarize), act in Gmail when you ask it to (archive, reply, report spam, unsubscribe, label), manage your Gmail filters — that is how a "silence this sender" rule is made to stick across all your devices — read and manage your Calendar including your free/busy times, use your Contacts to reach the right people, and keep the reminders it tracks for you in Google Tasks.
Authorizing something is not the same as it happening. These are the permissions Lonzo needs to do what you ask; what it actually does on its own is a much shorter list, and you can see and change it in your settings.
It all happens inside your own Google account under your own identity — Lonzo doesn't create a separate inbox or calendar, and your Google account stays the source of truth. It won't send to unengaged incoming mail without your instruction, and it won't act on sensitive matters without asking you first. How your data is handled is governed by the Privacy Policy and Google API Services Limited Use Disclosure.
See also: Does it act as me? · Where does my data live?