In an underlined word, how can increase the space/distance between the underline and the word itself? How can I further stylize the underline (for example, making the underline bold)?
I don’t think you can do that even in InDesign. 
You could look at Format > Font and Format > Paragraph menu items for what’s possible in Scrivener…
Even if you export to Word it may or may not work (even though they have a setting for it … depends on the font).