Does an IDA Pro Essential license connect to your cloud server when decompilers are not in use?

No.

However, if the user enabled automatic Lumina server usage, IDA might try to fetch metadata from lumina.hex-rays.com at the end of the analysis. This can be turned off in settings (or just not enabled when asked on the first run). 

IDA can perform periodic version update checks (once per week).

They can be turned off in the Help > Check for free update... dialog.