Hey David -
1. You're not doing anything wrong. This used to work. Tableau Public just took an update, so you may have found a bug. Thanks, I'll pass this along.
2. You cannot turn this stuff off. Tableau Public is "Public" and we want people to share / de-construct other people's work and data. (therefore the Download button). Same thing for the Tableau logo. This FAQ will probably address most of your questions:
FYI, there is a way to buy / upgrade to a "professional" account which will allow you to turn off Downloading.
Hi David, regarding #1, is it necessary to use a local html file? It looks like Tableau is taking the protocol from the base URL (file://) and using it for all the other URLs (in place of https://) and this is failing.
I don't know if this change is intentional or not, you'll have to wait for Russell for more information. (If I had to venture a guess, I'd say it was intended, as it's a common technique to convert embedded http references to https when the base page is https.)
This issue is fixed in a later version of Tableau Public and Tableau Online. We won't see it publicly probably until the next update to Tableau Public/Online though.
In the meantime, if you throw your HTML file onto a web server, it won't be a problem. It's only an issue if you view it locally (i.e. opening the .html file directly from your computer).