Answer: Yes, you can form one-person C corporation under Delaware's laws.
In the U.S., corporations are created by statute and every state statutes that regulate their formation and operation.
In Delaware, you are allowed to have one-person corporations. All corporations have three tiers of participants: the shareholders, who elect members to the board of directors; the board, which in turn appoints officers to manage the company on a day-to-day basis; and at least three corporate officers: president or chief executive officer, treasurer or chief financial officer, and secretary.
This structure applies equally to the single shareholder corporation. It is just a matter of having the proper documents and paperwork. Something you want to ensure gets done correctly. You would need the assistance of a person or company–much like the Stripe Atlas Service you reference in your question–to act as your registered agent. A registered agent is a business or individual designated to receive service of process when a business entity is a party in a legal action such as a lawsuit, summons, or subpoena. Basically, they act as your business's presence in the state of Delaware.
- The corporation must issue a stock certificate to you showing that
you are the shareholder.
- You must prepare minutes from a meeting where you, as the shareholder, elect yourself as the sole director to the board.
- You then must prepare another set of minutes covering board of director matters, which would include showing that you, as director,
have appointed yourself to the three officer positions.
Additionally, you may want a lawyer to ensure everything is set up correctly because the consequences of not following the corporate formalities can be severe. Failing to follow the requirements for electing a board of directors and appointing officers would cause someone to be able to pierce the corporate veil. If someone pierces the corporate veil, that means you lose one of the main benefits of operating as a corporation: limited liability. Without the limited liability in place, someone could go after you personally for the company's debts or liabilities. (This probably isn't an immediate concern since you are in Russia, but is something that you nevertheless should keep in mind.)
Also, the C corporation part is something that has nothing to do with how many shareholders, board members, or officers are in your corporation. It is a tax issue. By default, your corporation would be a C-Corp. As you already figured out, because you are not a U.S. resident, I think a C-Corp. is the only type of for-profit corporation you could own as far as the I.R.S. is concerned.