Finding last reboot time:
# last reboot
# last -F
# who -b
Finding last shutdown time:
# last -x | grep '^shutdown'
Listing out failed login attempts:
# lastb
The other straight forward way to find the failed login attempts is to analyze the /var/log/secure file. I have written a Perl script exclusively to report the failed login attempts by parsing the /var/log/secure file.
Get the script from here: http://smart-scripts-for-sysadmins.blogspot.com/2012/05/perl-script-to-list-failed-login.html