If I change a Field Name, would it be updated on existing tickets?

Yes, changing the Ticket Field Name would automatically change the Label for existing tickets and the Field would also be automatically updated. Please navigate to Admin > Workflows > Ticket Fields click on the field to be taken to its properties window where this change can be made. 

However, in the case of drop-down fields, if one of the choices within the drop-down is deleted or edited, the value would not be updated on existing tickets. The value of the field, in such cases, would revert to Null(...).