What regulations apply to changing liveries on private jets in the US? Things like the manufacturer's livery, do they need to follow that?

Anything else that will be required when purchasing a new jet and changing the paint to the new owner's choice and requirements.
The only requirement is the height of the registration number, 12" if flying internationally (from the US anyway), 2" I think if staying domestic, and maybe new weight &amp; balance - but usually a layer of paint and primer is taken off, and a new layer is put on, so the weight change is minimal.  Any control surfaces that were removed may need rebalancing.  Colors &amp; schemes are at the whim of the owner(s).