Can Project Manager be an individual contributor role?

Yes this is possible for reasons for example where the project is very small considering the schedule and scope.It can be of individual contributor role.The major challenge will be the PM is the one who had to take the decisions and in control as he cannot fall back upon his team for discussions.

Yes, PM is responsible to manage balance in cost, time & quality to achieve Project goal. If you can control these factors even people not reporting to you, you can be effective PM, Only challenge with such situation people who are not reporting directly may not listen you and their reporting managers may not bother about your work. In such case PM needs to demonstrate his leadership qualities to get things done within a given environment.

I usually hold different roles for different projects: project manager, resource manager, team lead, business analyst, even enterprise architect.

In that last role, I was the sole contributor on the project. I made sure that I had someone else designated to "manage" me, even if she did not count as a project resource.

Yes, think about the example project of preparing yourself for the PMP exam. You are the project manager and probably only human resource. You may be even the sponsor. But you might have many stakeholders.
It is a project as it creates a unique result and is time limited (though many shift milestones).