1. Whatever begins to exist has a cause;
Defining a "cause" is more complex than that, because several factors could
be the cause. Many times, things have several causes.
True there are things that have multiple causes but that still means that they HAD A CAUSE.
We don't know if everything needs a cause.
That is why the Kalam only deals with things that begins to exists. It's not design to explain that everything needs a cause only those that were brought into existence.
Theists believe a god or gods exist without cause.
This is a concrete example of your complaint above. God is NECESSARY and ETERNAL and therefore there was never a moment in time, or outside of time, in which he was brought to existence.
2. The universe began to exist; and
This is what we currently think, but for a long time, scientists believed
the universe was eternal and infinite.
That was because there was no proof back then that the universe was expanding.
Then Hubble came along and discovered that stars and galaxies were moving away from each other which would mean that at some point in the distant past, they were very much closer than they were now and if you move even further into the past, you reach the singularity were time, space and matter began to appear.
It's equally possible the universe is both infinite and eternal.
No it's not. The evidence is leaning to a finite universe. The Big Bang and the BGV theorem shows that.
3. Therefore, the universe has a cause.
But god(s) exist without cause, so it breaks the rule.
Saying the rule can only be broken once, via God, is not following the
deductive reasoning.
If the universe began to exists during the Big Bang, then it would mean that the universe is CONTINGENT.
A Contingent object means that it depends on other causal forces or objects for its existence.
For example, you and I are contingent beings. We all need our parents to procreate in order for us to be born on this world. Likewise, our parents are also contingent beings because they need their parents and their parents and their parents etc...
So if we have this very, very long chain of contingent objects/beings, then if we trace it all the way to the beginning, we will soon find out that the first cause could never be a contingent object/being and thus it had to be NECESSARY.
Thus, this First Cause must be uncaused and not of this universe which means it's spaceless, timeless, immaterial and incredibly powerful (because it created the universe).
So again, you said this is not deductive reasoning, actually, if you follow deductive reasoning, you will definitely end up with the First Cause Uncause Prime Mover which theists would call God.