Do all arguments for the existence of God require a belief in the existence of abstract entities, aka universals?

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It’s less about God’s existence than about his definition.

What is God? If your definition requires his existence to have abstract entities, then yes.

Also there is the problem of what the difference between an abstract entity and a regular one. There is a class of things that does not exist, yet affects the world regardless. These are ideas. Let’s say I have an idea for a new kind of mousetrap. Mousetraps already exist, but not your new, better version. You write about your new version but nobody really listens. Your idea, which only really exists in your head, has affected the real world, writings about the idea exist.

This should feel problematic from a materialistic point of view. Do ideas exist, if so, then where? Does math exist, if so, where? Does your better mousetrap exist?

An idea sounds very suspiciously like an abstract entity. Math looks like it’s very definition. It sounds foolish to me to try to eliminate the non-physical. So forget God, you can’t even account for numbers without abstract entities.