As with hiring for any employee in Europe the first consideration must be to use a locally compliant employment contract, please do not under any circumstances consider that a US format employment contract will be sufficient. In the event of a future HR problem, trying to enforce terms for a German based employee engaged under US terms will be an automatic disaster and won’t hold any water from a legal perspective.
At a basic level the minimum statutory terms would be a default position in such circumstances and that would be a best case scenario that would be quite unlikely to be a final settlement.
Hiring German employees does not require a local subsidiary entity unless those employees work from a physical office with a lease held in the name of the company and, which would be rare, those employees were authorised to be the signatory to sales contracts between the company and its customers/clients. A salesperson facilitating or negotiating sales deals does not generally count as an authorised signatory from this standpoint.
Employees hired under a non permanent establishment in Germany are subject to a different taxation process from those employed in a German based entity. Social taxes in Germany are broadly split between the employer and the employee, with at the time of writing the employer contributing a little over 20% of the employee’s total pay.