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All relative date functions in Tableau are handled in a very special manner (read with care and love) so yes, TODAY() will always be evaluated at the viz rendering time. Even more, calculations based on TODAY() will also be recalculated so if you have something like YEAR(TODAY()) it'll do extra work every time you open workbook.
For the second part, I don't think it is currently possible to have only TODAY() selected on the filter. For that I'd recommend using relative datetime filter and configure it to select "last 1 day".
Might be of interest: https://www.dataplusscience.com/TableauTips10.html, tip #5
Another new feature in Tableau 10.3 that will be super handy. You can now filter based on the most recent discrete date. This will allow you to have data that can be updated at regular intervals (ex. daily or monthly) and your visualization will now open with the filter set to the most recent day in the data. Click on a discrete date filter and select the checkbox for "Filter to latest date value when workbook is opened".