I know feasibility studies are recommended by a number of project management methodologies i.e. The Waterfall Model, but would a feasibility study be method for project management?
I think the answer is, it can be. A feasibility study can be part of your overarching project approach and can be further described in your PM methods, in terms of the description of how you would do one, the outputs, timing, estimated costs, required roles and expertise, etc.
No and No
Have a look at the definition of a feasibility study (e.g. in Wikipedia): ‘A feasibility study is an evaluation of a proposal designed to determine the difficulty in carrying out a designated task.’ Hence a feasibility study can be part of a project. Feasibility studies are not part of a methodology. But some methodologies give advice on how to handle projects which have feasibility studies as they introduce a higher level of uncertainty and uncertainty makes planning more difficult.
No, it is a decision making tool.