Pe, Pi, Je and Ji ?????? Where did you get that from? I only know Jung and MBTI: P and J are not functions, they are indicators to which functions are stronger: P=S or N dominance, J=T or F dominance
You are correct in your understanding that there is no cognitive function called, say "Extraverted Perceiving (Pe)". However, it can still be useful to use the term "Pe" to be a place holder for Ne/Se. When I use "Pe/Pi/Je/Ji" I am using them as place holders for the functions that fill their slots.
Also, to give a more well-rounded answer in general to all your questions:
Everyone needs all four of what I call the "flavours" or "languages": N; S; T; and F. This is because everyone needs a way to be interpretive and literal, and to use logic and values. If someone lacked any or multiple of these at their conscious disposal, they wouldn't be able to deal with life.
Everyone also needs all four of what I call the "roles": Pe; Pi; Je; and Ji. They are all required to fulfil four sets of critical roles.
There are a total of 4 combinations of (four) functions which fit both these criteria: Ne + Si + Te + Fi; Ne + Si + Fe + Ti; Se + Ni + Te + Fi; and Se + Ni + Fe + Ti.
Nature has found a way to specialize humans by granting us the cognitive function hierarchies - they determine: which functions each of us have at our conscious disposal; which give us mental energy; which drain us; and in what order of proportions (ratios) do they give/drain our mental energy. These hierarchies have been interpreted by many humans as "personality types".
(The hierarchies do not determine how well developed or powerful one's cognitive functions are; they just determine how much stimulation (or lack of) each function gives one.)
The first function in each hierarchy is the dominate - its use grants the most mental energy for the user. The second function is the auxiliary - its use grants less than the dominate, but still some mental energy for the user. The most efficient/functional combination for the dominate and auxiliary is that they are of opposite attitude (I vs E), and of opposite (ir)rationality (P vs J) - this is because with a two function combination like that, one can deal with challenges that require: an introverted function; an extraverted function; a perceiving function; and/or a judging function.
However, some challenges in life require more specific cognitive function classifications; e.g: that a N, S, T or F function must be used, or a Pe, Pi, Je or Ji function must be used. For this reason, nature has granted each of us two additional functions, but these drain us when used. These are: the tertiary, which drains us a bit; and the inferior, which drains us the most.
It is true that it would have been best if the tertiary was of the same attitude as the auxiliary, and the inferior was the same attitude as the dominate, for then the functions would go: EIIE or IEEI - the best balances. However, an equally or more important thing to be balanced is the (ir)rationalities: JPPJ or PJJP, which also go hand-in-hand with the "flavours" (N/S/T/F), together making them defiantly the higher consideration. The EIIE/IEEI ideal is not compatible with the JPPJ/PJJP ideal, for if they were both applied by nature, then one wouldn't get a conscious arsenal of all the four roles (each person would then have two pairs of Pe/Pi/Je/Ji and lack the two others). Thus, the EIIE/IEEI ideal takes one for the team and becomes EIEI/IEIE to accommodate the four roles.
Combining the JPPJ/PJJP and EIEI/IEIE principles, we have: Je-Pi-Pe-Ji; Ji-Pe-Pi-Je; Pe-Ji-Je-Pi; and Pi-Je-Ji-Pe as the four allowable arrangements. There is thus four allowable hierarchies per each of the 4 combinations of (four) functions, making 16 types.