Recently had to troubleshoot a Intune Exchange On-Premises Connector issue. Here are my takeaways:
In my case the issue was that the Global Admin Account used for the Connector configuration needs a Intune License assigned (not documented as far as I could tell).
Information for Proxy settings (From 2015 but still correct):
https://blogs.msdn.microsoft.com/beanexpert/2015/06/18/microsoft-intune-exchange-on-premises-connect...
Information for Logs and Verbose Tracing:
http://ilantz.com/2015/10/23/intune-on-premises-exchange-connector-log/
Official Microsoft Troubleshooting steps:
https://docs.microsoft.com/en-us/intune-classic/Troubleshoot/troubleshoot-exchange-connector
