Event title's line spacing too tight

You're guess about adding some custom CSS is correct. You're site's default for spacing actually is only about 1 pixel more than EventBooking's value, so you could have this issue other places on your site as well with long titles if other components don't reset the h1 tag's line-height property to a greater value.

Anyhow, to fix this for your event listings, go to [ Event Bookjng -> Configuration -> Custom CSS tab -> Custom Css text area ] and enter:
I thought 2em looked good on my browser. Adjust the value if the spacing isn't quite right for what you want to see though (eg. 2.2em, 1.8em, etc;or you can use pixels as the unit of measure eg 24px, 18px, etc.).

As for the box, it might not be a part of Event Booking????

Looking at the code , it is ID'd as "pop-print" and the link looks like it is supposed to call a template-less version of the current page. It doesn't seem to do anything on-click however.
You can make it hide whenever an EB page is displayed by adding this CSS to EB's custom css box as well: If this still shows up on other parts of your site, you could add the same CSS code to your template's custom CSS (if it has an option for adding your own CSS ).