There are two separate issues here, I believe. Your reading of the simulation hypothesis is unorthodox, usually what it is taken to mean is that our world, with all of its laws and objects, is run like a videogame on a "computer" in a world, whose laws may potentially be completely different. Because of that it may be pointless to ask "how" the simulation runs, the answer might be unintelligible to us. But for what it is worth, it is usually imagined that the simulation "runs" constantly, so its inhabitants do not blink in and out of existence. There is however a gulf between the full-blooded existence supporters ascribe even to simulated critters, and the kind of rarefied pseudo-existence that Mandelbrot sets have, see How can the physical world be an abstract mathematical structure? So I do not think that saying that simulation is not really a simulation because it "pre-exists" as an abstract something, works. This is not to say that it is not objectionable on other grounds, see If we live in a simulated world, doesn't there have to be a first world that's real?
But the simulation hypothesis is not vulnerable to the particular objection you raise also for another reason. "If I create a program that renders a Mandelbrot fractal, I did not invent the Mandelbrot fractal, nor did anyone else" would have made sense at the time of Plato (if he knew of fractals), but not today. Few now believe in a Platonic repository of mathematical truths and curiosities. What is invented can be declared pre-existent and treated as if it were discovered. It is a practically useful habit to do so, but nothing more than that, "some sort of abstract plane or definition" are just figments.
Arithmetic was most certainly invented by people, albeit not without the nudging from the external world. What it would mean that it "pre-existed" is not entirely clear. It may seem that in mathematics something is already there once the rule for it is "laid down". But even that is somewhat controversial. As Wittgenstein put it, when it comes to doing arithmetic God has no principled advantage over us, even he can not count without actually counting:
"Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us. We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No."
Tait explains it further:
"Concepts such as that of number, like rules, rest finally on our disposition to use words. There is nothing behind this usage, e.g., mental or physical states or some hidden mathematical reality, against which it can be measured for correctness. We can agree now to use the number concept in a certain way... Not only is our agreement concerning the number concept not binding on future generations, but our agreement is spelled out in ordinary English (ultimately) and depends on our common disposition in the use of that language. A sufficient breakdown of those dispositions would render the question of "same number concept" meaningless."
