Re: AJA Kona LHe...What does it do exactly?

You have a sync reference issue. The Kona and the monitor need a common video source as reference as does the Beta machine if you are playing out to tape or capturing. The easiest thing is to switch the Sony monitor off external reference and it will automatically sync to the incoming signal from the Kona.

It sounds like you have a complicated setup which isn't properly synced. Best bet would be to get a tech who understands all this to look at the setup, connect things appropriately and show you the options in the Kona control panel to make sure you are properly monitoring.