Managing log files (specifically for when they don't come in)

I apologize if there is a relevant Post regarding this topic.   I had a look through - and don't see any.  

We have about 40 - 50 clients running this software on workstations, servers, and small file servers running Win7pro.  

I have all the logs being sent to a specific email address, and some go to the clients as well.  

What I would like to know is if there is a good way to manage these logs other then manually reading all the emails.  

Is there a log program that could take incoming email logs, but tell you if it didn't receive one?  For example: a backup volume fills up, the backup stalls, and it doesn't send out a log.  The backup wlil stay "hung up"  until someone happens to notice that we aren't getting the emails.   I would like to have it alert me when there isn't a log file.  

2nd request:  to receive an alert when the email log file has certain keywords like "not enough space", or "did not work" - so that i know to respond.  

it gets tiring looking through these logs because most of the clients do not pay us to monitor it for them, and don't want to take the time to deal with it themselves.  

Thanks!