Underline: How do we get the spacing larger away from the words?

As far as I know this only works in TextEdit, by raising the baseline several notches (Format > Font > Baseline > Raise). Although the same command is available in Pages, it doesn’t act the same.

Anyway, instead of the character formatting better use graphical elements (Insert > Shape > Line).

It's my understanding that the underline position and thickness is specified in the font by the font designer. You might experiment with other fonts to see if another looks better.
Aside from that, I think you have to go with the previous suggestion of placing a line object in the appropriate location.

You can add a line above or below a word ( or both) and apply an offset. Under the text inspector>>more
There is a bug though where it wont accept a number added to the offset and you need to click the stepper up or down.

But you probably wanted a double line under the text which i dont think you can do.

I think the previous posting was about using paragraph separators, which extend the entire width of the paragraph (as well as being a single line).

Pretty much every font I've tried has the underline well below the baseline.
With double underlines it sometimes touches the baseline, but still only intersects decenders (like y and g).

Are you seeing something different (other than intersecting descenders), if so, what fonts?

Here's a trick that might let you tweak the double underline position. Change font of the space following the word you have double-underlined to a different font, one whose underline position is lower, like Blackmoor LET, for example.

Change font of the space following the word you have
double-underlined to a different font, one whose
underline position is lower, like Blackmoor LET, for
example.

Afterwards, if you increase the font size of Blackmoor LET, the double underline will go down even further. If you don't like the way it messes with your overall line height, you could put whatever you're (double-)underlining into a text box, resize, and place that on top of the desired position.

.

+the underline … only intersects decenders (like y and g)+

Apart from that Mrs Lincoln, how did you enjoy the play? (Tom Lehrer)