First of all, this is really good work! Looks great. I'm surprised that you're new to Tableau.

Adding these dots is tricky because that would require use of a dual axis on rows, but you're already using that. Let's pretend for a moment that you're not using it for the label. You could create a calculated field like this:

Dot

// Measure just for these three years so we can plot dots.

IF [Year]=2007 or [Year]=2008 or [Year]=2009 THEN

    [Below Minimum Wage]

END

Then use that as a dual axis.

But you also need the labels for each chart, which makes this tricky. Still thinking about how I'd approach it but wanted to get some thoughts down.

Could use a reference band like this:

Here's another idea. I don't love it, but it may work. If you're using 2018.3, the you can leverage transparent sheets. Create a copy of the sheet, then on that copy, make the line color 0% opaque (i.e. transparent), make all the axis header text and title white (to match the background), and turn off all the lines. It should look like this:

Then, on the original sheet, change it to be the dotted line like I showed before:

Now, on the dashboard, drag the label sheet onto the dashboard. Next float the chart sheet on top of it. Using the Layout panel change the x and y positions to 0 and make the width and height the same as the dashboard overall. Finally, right click on that sheet, choose format and change the background to None.

The result will look like this:

It's hacky but might work.