How do I set up a line going through text, and is it possible to set it up as a style?

Set  your first part with a Tab and then add a tab leader of a hyphen

Select the tab and reduce the tracking so it becomes 1 connected line

Create a Character Style based on this

After the text - add a Right Indent Tab

Insert a Hyphen into the Tab Leader

Apply the Character style you just created for the line - it should be a connected line now

The right indent tab will force it to the far right of the text frame each time. 

In your paragraph style

Go to your Nested Style section

Select your Character Style for the line - and choose Through 1 Tab

None up to and insert this character ^y in the section (you can type into it)

Character Style for line - up to - End of Nested Style 

Apply it to your other headers

To trigger it - you need to insert a Tab before the text and a Right Indent Tab after the text.

You need ParaStyle:

And CharStyle applied to the words alone: