Thanks for the response
You will need to contact HMRC Employers team to set up a direct payment scheme, which will allow this individuals salary to be paid through PAYE (also now RTI - Real Time information) so the tax and National Insurance are deducted appropriately.
The contact details are
And the employers name and address will be the same as the employee himself, as in essence he becomes a branch within the UK for the USA company.
Then you just operate PAYE as normal, and as a UK individual they certainly would have to meet UK employment law requirements
I have also added a link for the holiday pay position
Holiday entitlement https://www.gov.uk/holiday-entitlement-rights/entitlement
But as for the rest, there is an employment law forum within Just Answer, so do post a further question there and they will be able to assist you.
If you have any follow up questions regarding the tax side of matters, then please do ask, and if in the meantime, I could trouble you to rate the level of service I have provided, it would be appreciated, as this ensures Just Answer pay me !