I am not sure I fully understand your question (display of buttons 'at the same time').
You can always override the component's code to add bootstrap code into it.

Thanks for the clarification.

Well, this is the Joomla's behavior, the component does not create the pagination, but Joomla does. When there is no previous page (for instance), the previous link does not appear in Joomla.

Indeed.
BTW, I saw that your icons in the events' page are incorrect. You are using the 'last' icon as the 'next' and vice versa. Same for 'first'/'previous'.