On a Checkbox formula field, you do not need to explicitly set its value to True within the formula - all you need is for the expression itself to evaluate to True.
For example, instead of using the IF statement, just start with Case_Owner_Profile__c <> 'Field Service'.
When that simple statement evaluates to True, the checkbox should be shown as ...
There are only a handful of formula functions that are supported by encrypted fields. Check the list here
Supported operators and functions are below
& and + (concatenate)
Also carefully note below
Formulas can return data only in text, date, or date/time formats.
For your use case, LEN is not ...
Use functions MONTH and DAY in the formula, to check whether Live_Date__c month and day matches the criteria you are looking for.
In this way, you can keep it generic irrespective of the year in the date.
This formula returns true if Live_Date__c falls between 12/15 and 02/15, this can help you to proceed further :
(MONTH( Live_Date__c) = 12 && DAY( ...