How to add class or id to shortcode HTML elements?

You could abandon the shortcodes and use HTML.

<div class="row full-width whatever">
[one_third]1[/one_third]
[one_third]2[/one_third]
[one_third]3[/one_third]
</div>

OR

you could easily just wrap all of your code in another div and target the [row] element through CSS:

<div class="your_special_wrapper">
[row]
[one_third]1[/one_third]
[one_third]2[/one_third]
[one_third]3[/one_third]
[/row]
</div>
<style>
.your_special_wrapper div{ /*whatever css you need*/ }
</style>