I was wondering if I import shapefiles with unknown spatial reference into a feature dataset, with desired coordinate system, would that work?
No, it will not work (unless you're just using the import to define the CRS to what you know it is supposed to be). When importing to a feature dataset, everything must end up in the same CRS. The import process can automatically project from the source CRS to the feature dataset's, but in order to do that it has to know the source CRS in the first place. If it doesn't, because it's defined incorrectly or not at all, it will essentially define the source as target and make no projection. This is just like using the Define tool and setting it to whatever.
If you have an undefined coordinate system, you must properly identify it first and then define it before the software will be able to do anything with it automatically (at least, correctly). It can't project if it doesn't know what it's projecting from, or it will give you bad coordinates if you tell it to project from something incorrect (which is even worse than just defining it incorrectly since you'd be transforming values).
If you know what CRS they should be in, it sounds like you're just looking for a way to batch project, as described at Defining projections for multiple shapefiles in ArcMap? If you don't know, or they differ, you're going to have to investigate them all individually.