Generally yes. But not always. Sometimes there's only a single such object. For example, there's a largest sporadic simple group. It is a very interesting object. But there's only one of it.

To use a slightly less silly example, up to isomorphism there's only one ordered complete archimedean field. We call it R and we care a lot about it.

Also, sometimes you lack enough data to know if there are other examples of what you care about. But yes, you should generally try to figure out if a non-trivial example exists before you start studying it.

Not when the question is whether any examples exist!

Non-Euclidean geometries? IIRC the questions of "what can you still/now prove with this one postulate removed" were studied for centuries before hyperbolic or elliptic geometries were really understood.

Or maybe I'm misremembering. That always did seem odd to me. I guess hyperbolic geometries can't be isometrically embedded in R^3, which makes them hard to intuitively comprehend. But the educated classes have known the Earth was a sphere for millennia; surely somebody noticed that this was an example of an otherwise well-behaved geometry where straight lines always intersect.