Largely you can't; while Anglosphere jurisdictions generally have easy company formation, it can get increasingly difficult and complicated in other countries, and there's usually an ongoing filing effort to maintain the company presence.

Some countries like China simply won't let you incorporate there without a local "partner".

Same with India. You need a "local" Director.

This is basically one of the reasons large outsourcing firms exist.

No, but there are companies that are already incorporated in many countries across the world, and you can then use to employ people (ie you pay them, they pay employees in the target country).

They are called PEOs or EoR (professional employment organization / employer of record.. There are differences between the two)

When I've had this discussion, apparently not. Even within the US, it's not as simple as just "have ADP handle it" because lawyers need to be involved and there are even differences among states on how PTO needs to be handled for example.