Hi Amaury,
Regarding #1, I had a quick look and it seems to me it has not been noticed that this particular formatting option is not working for a map visualization. (This may because it can be difficult to display different dates at the same time on a map that can only visualize one data point per map element.) We have entered this issue to investigate and fix with you as a submitter so that we will notify you of its progress and I think you can track it from your account page on the support site.
Option #2 should still work, though, because the formatting used by a time dimension is up to you. Are you sure this specific metric set used on the map is using a time dimension? If it is currently displaying a format similar to MM/dd/yyyy HH:mm:ss, these are either the original or ‘raw’ dates before being replaced by a time dimension or else it would have to be one of the Seconds levels of a time dimension that has been customized to include those levels.
You can either edit one of the existing time dimensions in the Time Dimensions folder in the Explore window, or you can create a new one just for this map and then edit it after saving it. Expand Formatting, change Hierarchy Level to Day (this corresponds with your format) and set the Caption Format. If you were already using this time dimension in a map that is currently on the screen, you would need to click to Refresh Data or reopen the file/page for it to reload its data with the new format. Otherwise, if you expand a time dimension in the Explore window all the way down, you can find the exact level you want to display (e.g. “Day”) and drag that on top of your “date” in the Data Analysis Panel, or if you are using a data cube you would need to edit its Process Result to replace your dates with the time dimension hierarchy. Once the time dimension is being used, its formatting is used by default everywhere from tables to map tooltips. Let us know if you are still stuck!
