I am based in Germany and would be required to pay my taxes in Germany (?). Going with Stripe Atlas, would I have to pay additional tax in the US as well?
Whether it is "additional" tax depends on your circumstances, and is a good question to run by an accountant.
I've run US businesses from Japan for ~10 years or so. The net tax rate of doing is not obviously materially higher than just having a Japanese entity and doing everything through it; I'll avoid giving you the full blow-by-blow of why that is true, since it isn't relevant to your circumstances, but the short version is "foreign tax credits" and "social security totalization agreement " (a bilateral treaty).
I would love to give you an answer regarding the Swedish tax system, but it is not one of the two tax systems that I know enough about to model well and, unfortunately, if it were, I would be legally enjoined from doing any sort of interpretation about how our product interacted with it.
I recommend you talk to a local accountant.
The company would then be liable for German taxes on income generated from Germany. German Payroll taxes would also be required if they employ people within Germany. I am unsure if the Branch Office is also required to pay IHK fees but that may also be the case.
Atlas makes more sense for people living in countries where Stripe is not supported but still want to use Stripe, the perk is, you have to pay high U.S. taxes that you won't enjoy and you will still have to pay German taxes from your U.S. income because you are a German citizen?