Can I use installed apps from unknown source even if I turn off "Install from unknown sources"? I have Moto G running Android 5.1. I had to turn on "Install from unknown sources" to install some apps from my employer.

Now can I turn "Install from unknown sources" off to be safer and still what I installed before from my employer will keep working without any problem in all cases? Also if I do the same in my new Samsung device running on Android 6.0.1?
Yes
Because enabling the option install from other sources , turns off the security checks carried out by Google Play  during installing only. It has no bearing on running the apps thereafter. So you can safely disable that and run the employer apps without any problem. This is true for all versions of Android ( I haven't come across anything to contrary in Nougat)
Download from sources other than Google Play says :

You can let your device download apps from sources other than Google Play by turning on the &quot;Unknown sources&quot; feature.
IMPORTANT: Your phone and personal data are vulnerable to attack by apps from unknown sources. If you download apps from unknown sources, be aware that your phone may be damaged or lose data.

There is no mention of running apps but you can turn app verification on to:

The &quot;Verify apps&quot; feature regularly checks activity on your device and prevents or warns you about potential harm.