Alright...
There is no explicit GMaster variable. On the client side there is a variable in network called @gmaster and it's set properly by the server. The server extension has a hash called @gmasters where it keeps track of all GMasters assigned for every map and a hash called @capable_clients where it keeps track of clients that can potentially be GMasters in case the current one gets disconnected.
When somebody connects, the message code MEN is sent (read the documentation for more information) and if there is no GMaster in the new map yet, the current one will be assigned.
The log is exactly what it says, it's just a log. If you turn it off, it won't write to a file what is going on makes it almost impossible to track errors and fix errors, but the server runs fine without logs.
Now, as for all your problems. If some unexpected error happens, the extension will crash. The thing is that it won't just crash, it will also shut down the server. That isn't happening with you which means that your host machine is somehow messed up badly. That's why enemies suddenly stop working.
The RMX-OS scripts work fine. You've said yourself that it works completely fine on your local machine. Only when you put it up on vlexo, you start getting problems. The problem is obvious, vlexo is causing you trouble. As far as I can see (from the 2-3 error messages you gave me), vlexo cuts off traffic as soon as the data string is a bit longer (which is ridiculous considering that the data strings are no more than a couple of hundreds of bytes) and refuses to write logs files (possibly because of file permissions) which you fixed by removing logging completely. It simply cuts off the message. RMX-OS cannot function if messages get cut off like that. RMX-OS is using a TCP connection which means either the message arrives as a whole or not at all. vlexo does not act as it is supposed to, it does not do it properly.
I think that the server extension crashes silently out of the exact same reason. The GMaster tries to send data about enemies to the other clients and because the string is too long, the server crashes in the extension code without properly shutting down the server.
I don't think I can do any more for you. vlexo is your problem, not RMX-OS. You should try to find another server to see how it behaves, possibly a Windows machine. You can even try to run it temporarily on your machine and ask a friend to connect remotely to your PC. This will show you that RMX-OS works fine for the intended environment which is a properly set up server on a Windows machine.