I would like to add a colored wiggly underline to certain text fragments of a element. The effect would be similar to that of spell checkers.>
As you say it is actually not possible to just edit the font or properties of one line of a textarea.
Instead, what the website you have linked to is doing, it using an editable DIV, and linking this through Javascript and some clever CSS to the hidden textarea (the textarea itself is display:none and visibility:hidden;)
I suggest using firebug to have a look at how they've done it, as it's pretty impressive.