The "Convert" feature creates a new Opportunity, Case, Lead etc from the tracked email, and sets the email as "Regarding" that new record. While you can enable other entities to have activities attached to them, this won't add any special buttons to "Convert" emails to records of these entities. But you would see the new entity appear in the list of "Regarding" entities available to choose.
You cannot enable an activity to have activities attached to it. You can add a custom relationship, from Appointment to EMail for example, but that won't use the "Regarding" lookup.
Best option here would be to use an on-demand workflow for the email record. User would track the email, then go to run workflow > select the workflow for Appointment or Task and confirm. Workflow would create the new record, copying relevant fields from the email.