Relationship Status Inquiry: As I am recently engaged to my fiancée who lives abroad, should I list my relationship status as 'married' or 'unmarried partner' if we are still engaged at the time of application?
You are not married, so you cannot state that.
Fiancée Visa Concern: If I plan to sponsor her for a visa and it gets refused, will that affect my Citizenship application, considering she will be living abroad. Also, is it recommended to sposnor her after I get Citizenship first?
No, this has absolutely no effect on your application. Why would it? It doesn't matter when you sponsor her. You are overthinking.
Travel History Details: When documenting travel history, if I visited two countries while outside the UK, should I list only one country and specify the travel dates for each country in the reasons section?
Should be fine.
Proof of living in the UK: As I have ILR, is my passport the only document required for submission to prove that I have been living in the UK or for example I need to add also council tax etc?
Passport only.