You've asked some interesting and illuminating questions... let me try to address them is a relatively short space... (I am not, by the way, a Hebrew scholar).
Firstly, as I'm sure you know, Hebrew, historically is a Semitically derived language and shares some of it's derivation with other Semitic languages, such as Arabic and in it's Proto forms may be as much as 5,000 years old.. It was also one of the first languages to develop symbology by way of an alphabet to express ideas using symbols. Over it's history, it came to be Hebrew by way of descendancy from the Cannanites (to lengthy to discuss here).
Having said that as a basis from which to develop your answer... the word or appleation EL, was apparently, originally a Ugaritic term meaning god...
It had other important meanings by the time Hebrew was developed... such as false god, mighty, power, etc. In Hebrew, when referring to the unspeakable name of the God of Israel, it isalways used in a form nominated by linguists as construct. An example is El Emet (sometimes seens as El Emhet)... God of Truth (Ps: 31:5) as well as your example. It is most certainly not short for Elohim... Elohyim (properly) is the plural of Eloah (-im always pluralizes nouns in Hebrew).
By the way "Eloi, Eloi, lama sabachthaniI" as uttered by Yeshua on the Cross, is Aramaic in origin, (as noted by the text), related to Hebrew, of course, but Elohim (Eloha) does not derive from Eloi, according to Strong's. Check Strong's 1682, H430 and H433.
Contd.