Unable to start the mongo services on debian server
That first log file tells that "client" connects to the server and gives dropDatabase command, so check your application. Why it executes dropDatabase.

What comes to the second problem, you need to check your mongodb.log, what it tells for the reason, why mongodb cannot start. Wild guess is ownership of some files (data, lock, log), all files must be owned by user what you use to start mongod process (probably mongodb -user)
Name and location of mongodb.log, you can find from /etc/mongod.conf -file