(1) is relative if you want to press it in that direction but (2) applies to the spacetime universe.
But I don't think we need to press (1) in an extreme direction. Would you admit that your person came into existence at a certain point in time?
Either you're using the phrase "began to exist" the same way in all of your premises and applying it the same way to all the things that "began to exist", or your argument is not valid. If you mean something different by "began to exist" in premise 1 than what you mean by "began to exist" in premise 2, then using the same exact phrase would be dishonest.
You're using the exact same phrase because you mean the exact same thing and you're applying it the exact same way to all things, though, right?
So let's say I, the person, began to exist once all of my molecules gathered together and electricity started flowing in my brain. We can say, "Look! There was a cause for those molecules rearranging, therefore there is a cause for molecules rearranging even if we don't know what that cause is in some cases."
Then we consider the universe. If it's a rearranging of matter and energy, or if it's a rearranging of something more fundamental into the things we call matter and energy, then we can say, "Look! There was a cause for that rearrangement even if we don't know what that cause is in this case." However, if you mean "began to exist" as in "popped into existence where there was previously nothing" no one has ever seen that happen so there is no reason to apply causation that we witness in rearrangements to "poppings".
So if you're talking about things rearranging in premise 1, and then the universe popping into existence in premise 2, your argument is no longer valid because "began to exist" means two different things in the same argument. It should look more like this:
1) Everything that moves has a cause.
2) The universe popped into existence.
3) The universe has a cause.
That isn't valid.
Personally, I just think matter and energy and probably even spacetime is just some other more fundamental thing rearranged to look like the stuff we find ourselves in. So I would be fine if "began to exist" in your argument always meant "rearranged", but I don't think you mean it that way in premise 2.