Unfortunately you weren't far from the truth on that second part about Gmail showing ads in your Inbox.
https://insightinvasion.com/gmail-ads-users-frustrate-as-google-tests-ads/
Not sure if they are still testing this out on people. I know I never got put into the group of users they tested it on but I'm at a point now where I don't trust it won't be pushed onto people sooner rather than never.