As I said, context is everything.
1. I have no idea what is in the contract, what sort of contract it is or what it is for, but if it is a contract for service, or even a contract of service, the UK contractor will be taxed on whatever he receives in exchange for his service. This can be in cash or in kind. For what it is worth, I’d venture that the agreement is part of the contract whether in writing or not.
That said, and I don’t profess to be an expert, are you sure that a Delaware LLC has stock or stockholders? My limited understanding is that, although bodies corporate, they have members like a UK partnership.
2a. The UK contractor will pay income tax on the value of the shares. Nobody will be able to say if it has value or not without conducting a valuation. The market value of anything is whatever an unconnected fully informed person, trying to get the best deal, would pay for it.
2b. Not sure what disposal you are referring to. If the contractor sells any shares he acquires he will be liable to CGT. His base cost is the MV (the amount he was taxed on) when he acquired them.
The sale of the business and assets to the Delaware company may have tax consequences.
2c. The company is in the US so will have US tax on its profits. It’s important to get it right because there will be a myriad of pitfalls with double taxation given the different ways that the US and UK treat the profits.
You are not going to get the proper advice you need without sitting down with a professional to discuss your needs and reasons why you want this structure. Good advice us a commodity, and it comes with a significant price tag.