This is going to be a little long-winded, since your question isn't very specific, and the subject is rather complex.
Question2Answer uses PHP code to generate the HTML that is sent to the browser. For a question in the question list the HTML structure looks roughly like this:
The class attributes in that snippet refer to CSS classes that define how the respective element should be displayed or if it should be displayed at all.
To modify the layout of the question list you need to adjust the PHP and/or CSS code according to your needs. CSS is pretty flexible, and for many design changes CSS alone should suffice. However, in some cases re-arranging an element also requires putting that element in a different place in the HTML structure. In that case you need to change the PHP code too.
Changes to the PHP code are implemented in the file qa-theme.php in your theme folder, where you override one or more methods from the class qa_html_theme_base in the file qa-include/qa-theme-base.php. I'd say q_list_item() and q_item_main() are probably the most relevant ones for your scenario.
Changes to the CSS code are done by modifying the file qa-styles.css, also in your theme folder. Beware that there may be different styles for any given element in the CSS, depending on whether the page is being displayed on a desktop or mobile system, so you need to pick the right one for your change. It could for instance be that the designer of your theme wanted to avoid cluttering the limited screen space of mobile devices and thus chose to hide the avatar in mobile view.