A script to ping and then login via SSH, then drop the connection. With logging and sendmail

Sure.
Assuming that you have a file containing your hostnames called hostlist and that the same userid is capable of logging in to all servers -
#!/bin/ksh
USER=myuser
for SERVER in $(cat hostlist)
do
  ping -c1 -w1 $SERVER >/dev/null 2>&1
  RC=$?
   if [ $RC -eq 0 ] ; then
    # "Ping succeeded on $SERVER. Now trying ssh!"
    ssh $USER@$SERVER uname -a >/dev/null 2>&1
    RC=$?
     if [ $RC -eq 0 ] ; then :
      #  "ssh succeeded on $SERVER"
      else
       echo "ssh failed on $SERVER but PING succeeded" | mailx -s "SSH Failure on $SERVER!" recipient@domain.tld
     fi
   else
     echo "Ping failed on $SERVER - not trying SSH" | mailx -s "PING Failure on $SERVER!" recipient@domain.tld
  fi
done
exit    
Please note the colon ( : ) here - "if [ $RC -eq 0 ] ; then :  " - it's very important!
wmp

That's fantastic, how would I add logging to this also?

Many thanks.....excellent help!