I don't claim to be an expert on the functions so take my opinions with a grain of salt.
a) If you know what you feel but don't know why would that be Fi or Fe, or maybe both but it depends?
Knowing what you feel personally about things is Fi. Knowing why you feel that way sounds like a non-F function, maybe Ni or Ti?
I think I (respectfully) disagree somewhat here. I think any type can learn emotional awareness, but I agree that Fi tends to put a higher value on personal emotions/valuations, and Fi-ers tend to have more friendly and aware relationship with their emotions that most other types. For Fi-ers, an emotional reaction is valuable information: it points to something, even if the emotional reaction itself isn't "truth."
I think that Feeling functions naturally build up truths over time, and those truths include an understanding of emotional and social cause and effect. Being able to express those truths externally in a way comprehensible to others may require Thinking, since the interplay of subjective values and feelings can be hard to externalize clearly.
An aside on emotional awareness: just about anyone can learn how to be emotionally aware, and for some focusing explicitly on the bodily sensations that accompany emotions can be very effective. To me that approach tends to comes across as too mechanical and simplistic, but those who routinely ignore their emotional state tend to lose physical awareness along with emotionally awareness.
As aside about types and their relationship to emotion: It's been eye opening to me on this forum to hear about how others relate to their own emotions. For some younger INTPs, strong emotions seem to be experiences as an alien invader that will unexpectedly rise up and interfere with rational thought. Some INFJs tend to feel their emotions to powerful and dangerous to express directly, and that they should remain "the dragon in the basement". INFPs tend to view their emotions a continual barometer and as a companion that gives early warnings (dog or crying baby). Some INTJs tend to see their emotions as a vulnerability that is only shared with those they trust deeply.
b) If you think that e.g. you consider your mom strictly as mom, but not as a buddy/other person, is that Fe? (I'm not saying you have bad relationship with your mother for strictly considering her mom)
I don't see this as being Fi nor Fe. Putting things in categories according to preset criteria- Ti maybe?
Everyone is capable of seeing others as a person and not a role (okay, barring narcissists, sociopaths, etc), so I think one can only address tendencies. I think Fe tends to be more aware of social roles and attendant responsibilities. I think Fi tends to be less aware of social roles and responsibilities.
c) If you feel like you want to 'change' other people so they have your personal value system is that Fi or Fe?
Hmmm, tricky. The personal value system stems from Fi. I see Fe as more inclined to want to change others but I also see Fe as wanting to change others to conform to some external social norm rather than conforming to their own value system. Also I think Fi would also recognize that others have their own unique values independent of theirs. So I don't really know. Maybe a combination of the two? Or there might be some other functions at work too.
I agree. I think Fi-ers can lash out when a core value is trod upon (some things are just beyond the pale), but tend to leave a fair amount of room for differing opinions and outlook. This may not hold true, however, for very close friends and immediate familiar members who may feel silently judged for failure to live up to the Fi-ers values.
One way Fi-ers sometimes try to change others is to encourage people to "be true to themselves/their values." This can come across as relentlessly nudge-y in some circumstances.