Sound to me like the poster uses a virtual shared server, ie. The hosting company provides a virtual root environment, this would be an explanation why they can "root" the server, ie. just access the file system directly from the hypervisor.
I am sure this kind of service does not include uptime guarantees, it operates most likely under "Best Effort" promise. Which means they do not need to guarantee either uptime or availability. Their monitoring system and logs do not detect any error, so they want to check the posters systems logs - a reasonable request as they are trying to help him.
Monitoring is a difficult task - what to you measure, where is your sensor. Was the outage on your own internet connection rather than on the server?
"Moments ago, there were three simultaneous outages while I was logged into the server working on some projects. " It sounds very suspicious, like the admin interface causes interruption in the VM. Something like this happened to my VMWare Servers totally unexplainable, but reproducably.