Is there any function to display all application names in the environment.

No. The number of real applications does not directly correlate with the number of "Application" objects in Appian.

I suggest to check the documentation for available log files and their content.

For the recent activity of loggedInuser over time,  You can explore the use of Appian Usage Insights utility available on the Appian AppMarket. One of the metrics tracked is Logins by user (over time)

Hope this help!