Clearing the notification about battery issues
Based on the comment by Izzy above, I looked further under the Battery settings. There is nothing about Doze, but there is a section called Unmonitored Apps which apparently means "unrestricted" apps. I was able to add Outlook (but not Skype) to this list. The phone now says it will let them use all the power they want. Good.

Doing so has cleared the orange dot, which is great, and should also mean that I will get my push notifications. I guess. I don't really understand how Android works and seem to be constantly trying to stop it helping me.