Am I allowed to get into a Mutah marriage with a guy without my father's permission as he will never allow it being a non-Shia?
If your non Shia father refuses to allow your marriage with a good Believing person, for no valid reason, and you need the marriage to save yourself from falling in sin, you can refer to your Marja' of Taqleed to seek his guidance.
It is disliked for a virgin girl to do Mut'ah marriage.
Wassalam.