Do I need an Exchange Server?

You don't *need* one, technically you can manage all the related attributes just fine via the AD tools. However, the only *supported* method to manage Exchange-related objects and attributes are the Exchange tools, thus Microsoft's recommendation to spin up an Exchange box if you want to stay in *supported* configuration. I've worked with many organizations that simply ignore this recommendation, and they are doing just fine. But in general, using the Exchange tools is better, as they protect you from stamping invalid or duplicate attributes, etc.

As for extending the schema, this is pretty much a requirement if you are going to use AAD Connect - without it you will not be able to manage any of the Exchange attributes as they will be locked in O365 and not "known" at all in AD.