The built-in default properties of MixPanel such as $browser or $city all start with dollar signs. Does this have any purpose? Should we denote custom properties with dollar signs as well or is the dollar sign meant to be the marker for a default property?
It would not be recommended to start all of your custom properties with the same dollar convention. The dollar sign convention is meant to denote properties that the Mixpanel SDKs track automatically or properties that have some special meaning within Mixpanel itself. That link you shared is great for the default properties and there is also documentation for the reserved user profile properties and reserved event properties if you were curious about those.