What is the origin of the name Lexcorp?

Question

Okay, this may seem like an obvious question.

However, if I construe this properly, in Batman v. Superman: Dawn of Justice:

• The antagonist is Alexander Luthor, known as Lex
• He is the head of a huge corporation, Lexcorp
• Lexcorp was started by his father, who grew up in East Germany with nothing
• The father abused the son (see)

So:

1. Was Lexcorp always named Lexcorp?
2. What was Lex Luthor's father's name (also Lex?) ?
3. If the answers to the above are yes and not Lex, then does it mean that the father named the corporation after his son? Or his son after the corporation?

So essentially, in this movie continuity,

What is the origin of the name Lexcorp?

Answer 1

Although Jesse Eisenberg's character is officially credited as "Lex Luthor" at the end of the film, it was revealed prior to its release that his character is officially Alexander Luthor Junior (see Warner Bros. promotional article in Fortune Magazine).

This suggests that his father was Alexander Luthor. Also, in the film, Lex explains to the senator,

My father was the one who put the Lex in LexCorp.

The promotional article goes further:

As we patiently wait our turn at the complimentary LexCorp vegan food truck (this day’s fare: pesto-olive pizza with raw almond crust), the son of Alexander Luthor Sr. – Lex Luthor – explains the evolution of LexCorp.

"Dad named the company after himself ten years before I made my unexpected entrance into his life. But investors seemed to respond to the idea of an adoring father building a legacy for his precious son. He used that to his advantage. It was a good shtick and, whatever else he was, he was a good businessman," the younger Luthor explains.

So it seems that the father named the company after himself (and hence saw himself as "Lex"), but he revised history and allowed the public to interpret it as a gift from father to son in order to curry favour with investors.