Is there a way to display basic site statistics in a block?
This is a job for the views module. Using views in its most basic form you could have a block for the below in under 30 secs.  

Content statistics: Most recent view

The most recent time the node has been viewed.

Content statistics: Total views

The total number of times the node has been viewed.

Content statistics: Views today

The total number of times the node has been viewed today.

Install views module, add new view, view name = stats, uncheck create a page, check create a block, continue and edit. Add fields and add the 3 above. Remove title. Add a contextual filter under advanced and use Content: nid, provide default value = content ID from url. Get rid of any filter and sort criteria and you should be good to save the view. You will now have a view block that you can place in any region to display this info.

The 5 most visited block would be similar except you would keep the content published filter and add a sort (descending) on Total views. You could use the "Use pager: Display a specified number of items | 5 items" to get a list of the top 5 most viewed nodes.