Apologies for mistaking cause with explanation, but I would like to point out the similarity between the two:

Explanation - a reason or justification given for an action or belief
Cause - a person or thing that gives rise to an action, phenomenon, or condition

I would say they are slightly similar to say the least, but there is an important distinction, mainly between the abstract and practical. Because of linguistics and semantics and the topic we are discussing, this is a very important distinction. In theory, this helps you around the critical, "what caused the First Cause".

However, that is only because you turn the problem in itself instead of externally. You solve this internal problem by posing this clause:

I said everything needs an explanation of its existence.

Why? According to what? This is one of the larger holes in your argument. You assume things require an explanation in the first place. What makes it so? I believe you pose this as an "axiom" because that is how things around us occur. For example, my existence can be explained by an act between my parents.

I believe that is where your induction comes into play. "Everything in the real world requires an explanation for its existence, therefore everything requires an explanation for its existence". The problem is that we are discussing the metaphysical. You cannot rely on empirical inductions and then allude to the metaphysical.

You then bring forth God's explanation:

If a thing is self-existant, it exists of its own necessity, and must be eternal...
God's explanation would be self-existence.

How do you know when something is self-existent? I don't see an explanation for that. Can you explain "exists in it's own necessity"? If it is simply something it and of itself, I can argue the following:
Note how you described it. If you meant to use an if and only if relationship, as in **something is self-existent if and only if it necessitates itself and is eternal, that would not work. My counterexample is time. Time necessitates itself and is eternal because "eternity" is defined through time. That would mean time is self-existent. I'm not sure if time explains the existence of space or whether space is also self-existent. That would require more thinking.

eternal things don't necessarily need a cause, and self-existing things don't have causes.

It seems the concept of time fits your criteria to be God. In some senses, it fits better than God itself.

In the same line, why can't the Big Bang be self-existing?