Yes - you can view a single day by selecting it from the "Show day" dropdown box near the top of the right-hand column.
Yes, you can now view an individual quake. Pause the animation once the quake has appeared (alternatively, wait for it to finish by itself, or press the "go to end" button so that it lists all the quakes in the selected period). Then, when you hover over a quake in the list on the right-hand side, you'll see just that one quake (plus any visible "sticky dots"). Once you move your mouse off the list of quakes, the map will return to how it was, and you can click the play button to resume (or restart) the animation.
The US Geological Survey (USGS) provide global earthquake data in a variety of formats. This website picks out the earthquakes listed as being in Japan from these USGS global feeds.
The Japan Meteorological Agency (JMA) and the US Geological Survey (USGS) each gather their own data, using their own networks of instruments. This website uses the USGS data, as it's available in a format which can be used in this way - but the USGS data sometimes differs from the JMA data for the same earthquake.