The second part is easy--use the "Only Relevant Values" option when applying your filters.
The first part can be handled a variety of ways, depending on what you are trying to do. You could display both dates as continuous (green pill) filters, and when you set the range on both, only data that meets the criteria for both will be shown. You could also write custom calculations to create a filter or filters, or create parameters to set this up date filtering. A lot depends on your data source/structure, and what you are trying to do with the viz.
If you can post a packaged workbook that demonstrates what you are trying to do and the result you expect to see, we can try and help you set it up.
Here is a VERY simple example of the simplest form of two date filtering using the Superstore Extract data set. I limited the data to year 2013, and set two filters. Sales by Customer are shown only when falling within both date ranges. Again, this may not be what you want, so let us know more about the scenario and the data if this doesn't help.