1/r^2 has a nasty singularity at r=0, but it did not bother Newton-the Moon is far enough.

If I take the theory as we have it now, literally, I would conclude that extra dimensions really exist. They're part of nature. We don't really know how big they are yet, but we hope to explore that in various ways.

String theory is 21 st century physics that fell accidentally into the 20th century.

As far as extra dimensions are concerned, very tiny extra dimensions wouldn't be perceived in everyday life, just as atoms aren't: we see many atoms together but we don't see atoms individually.

1/r^2 has a nasty singularity at r=0, but it did not bother Newton-the Moon is far enough.

One very important aspect of string theory is definitely testable. That was the prediction of supersymmetry, which emerged from string theory in the early '70s.

As of now, string theorists have no explanation of why there are three large dimensions as well as time, and the other dimensions are microscopic. Proposals about that have been all over the map.