Hi,
As stated on the DNN download page, the minimum version of the DNN supported by Events 4.0.2 is DNN 4.5.5. I would think this is very likely to be the issue here. Version 4.0.2 requires Partial Rendering which DNN 4.3.7 does not support.
I would also think that it is very unlikley that the module will support an upgrade from 3.0.10. To understand why you have to look at this history of the module. The module was orginally written by Alan Vance as AVCalendar, which I suspect is what you have a version of. I believe the first version that was the DNN Events module is 3.1.0.
I think in order to get your data from AVCalendar, you would be better installing version 3.3.8, because that has functionality to support import from AVCalendar. I cannot comment how successful that would be because I have never tried it, and I came on board the team for version 4.x where this import functionality was removed.
Assuming a succesful import from AVCalendar 3.0.10 to Events 3.3.8, I would suggest you then upgrade your site to a current version of DNN (4.9.1), and upgrade the Events module to 4.0.2. Both the DNN and Events upgrades fix many bugs, and provide new functionality which I'm sure your site would benefit from.
Cheers
Roger