Is it possible to make a custom field and create additional customization options for when you add the column to a list?
If so, how?
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It is possible. It is another UserControl that implements an IFieldEditor interface. Once you have created the control and the field, you bind them together in the field definition xml.
Here is the step by step process https://msdn.microsoft.com/en-us/library/office/jj126861(v=office.14).aspx. Notice the FieldEditorUserControl in the fldtypes_InterestCalculatorField.xml file in the link above.