On the differences between El, Elohim, (l'havdil) elil, and elilim, what helped me to understand it was Reuven Hammer's commentary on I Chronicles 16:25-26:
כִּי גָדוֹל יְהוָה וּמְהֻלָּל, מְאֹד
.וְנוֹרָא הוּא עַל-כָּל-אֱלֹהִים
כִּי כָּל-אֱלֹהֵי הָעַמִּים, אֱלִילִים
.וַיי שָׁמַיִם עָשָׂה
Ki gadol Hashem um'huleil m'od
v'noreh hu al-kol elohim
Ki kol elohai ha'amim elilim,
va'Hashem shamayim asah.
Great is Hashem and worth of praise,
to be revered beyond all gods.
For all the pagan gods are mere idols,
but Hashem created the heavens.
Rabbi Hammer writes:
עַל-כָּל-אֱלֹהִים (beyond all gods) - Those that are worshiped as gods but do not exist.
אֱלִילִים ,וַיי (idols, but Hashem) - Although similar to the word אֱלֹ (el), God, אֱלִילִ (elil) has the opposite meaning: something unreal that is worshiped as if it were a god. When reciting this line, one pauses after elilim ("idols") before saying va'Hashem ("but Hashem"), to make certain that Hashem is not being identified as one of the idols.
To try to answer the rest of your question, "El" appears relatively rarely in Tanach. You can find it with Abraham in the name El Shaddai, presumably because he did not know the name Hashem (you know which name I mean without me spelling it) as that was not revealed until the time of Moses.
It is far more common to find "Elohim" which can be used as one of God's proper names (very often) or as a concept of "gods" (as in the excerpt above). I suspect that because of these two usages of "Elohim" (even not counting its usage as judges and rulers) and because we also have the proper name Hashem, means the the Tanach only rarely uses the shorter name "El".
It also might be noteworthy that "El" was a name that was used by the Canaanites for the highest god in their pantheon. That would be a reason that the prophets prefer to stay away from that term. Similarly, while one could theoretically call God "Baal" (master), we do not because that is a polytheistic Canaanite usage that the prophets worked hard for hundreds of years to get us away from.