Add the following CSS to your child theme's CSS or through a plugin like Reaktiv CSS Builder or My Custom CSS:
/* Hide archives header and title, comment icon on Event Espresso event list page */
.post-type-archive-espresso_events header.page-header, .post-type-archive-espresso_events .comments-link {display:none;}
/* Hide posted on date and author, navigation links on Event Espresso event list page */
.post-type-archive-espresso_events .entry-meta, .post-type-archive-espresso_events #nav-below {display:none;}
/* Hide posted on date and author, navigation links on Event Espresso single event pages */
.single-espresso_events footer.entry-meta, .single-espresso_events #nav-single {display:none;}
References:
http://eventespresso.com/wiki/hide-certain-elements-event-espresso-pages-using-css/
https://gist.github.com/lorenzocaum/e807be7a6d69d4cbc951