You sound more like an IxFJ then anything (I lean more towards INFJ). You show a number of examples of Fe at work. The reason being is you tend to adhear to social subjectivity on contexts external to you. If you were ruled by Fi this would be far less common, as Fi is largely internally driven. The feelings of others and the contexts of the social world are important to you. That is a sign of Fe, and a clear one at that.
You don't seem to give off too many examples of Si or Ni, so I am not inclined to say either N or S. However as you seem to test consistantly as an N, It is more likely you are ruled by Ni then. You stated that you like for things to be correct and right, which can be a sign of Si. However, you tend to refer to them in only social contexts, which could signify this manfiesting itself as Fe instead. I get the sense that you are more introverted then extroverted. The reason being is if you were a Fe dominant, you would not feel nearly as much hesitation with verbazling these convictions, and you would also do them much more directly.
You also said you tested strongly as INTP (to which I do not believe this to be your type). There is a signifigant minority of INFJ's who will type as INTP. This is largely due to a masking in a sense. Some INFJ's do not like to externally manifest their emotions, and instead only do this in a logical and factual way, by making them adhear tightly to social values. This causes their T score to go up as they would like their emotions to be logical and understood universally. However, this causes another shift in the P/J axis. Due to the inherient drive of INFJ's to hold onto fairness and uniformity within others, this will drop their J score. This is because when taking a test, the individual will compensate for their "harsher" T answers, for the "softer" P answers. Which could be interpted as being much more free flowing and allowing of their internal drive to be friendly and sensitive. Hence, it is possible for some INFJ's to test as an INTP. This is also the likely case for you.