Is there a way to find who visited the site at last and who downloaded file recently in SharePoint? Also can we log who are the users that used/accessed the file? Can anyone provide some guidance on how to do this?
Have you looked at Site Collection Auditing?
You can use the audit feature of Microsoft SharePoint Server 2010 to track which users have taken what actions on the sites, content types, lists, libraries, list items, and library files of site collections. Knowing who has done what with which information is critical for many business requirements, such as regulatory compliance and records management.
Otherwise, if you could go through the IIS logs for the SharePoint Web App and try to determine what you are looking for.