Most everything you said looked right to me, except you used z in place of t in the c2 term (I assume that's a typo) - and the metrics you use all describe flat spacial geometries. Different geometries emerge when non-diagonal components appear in the metric tensor/matrix in the presence of a gravitational field (things start getting complicated then) - Einstein's gravitational equation dictates how this occurs. Assuming that this 4th spacial dimension has a localized geometry, I'm making an educated guess that the measurable effects would be that gravity becomes much stronger at super-short distances. (What distance, I don't know - experiment rules out anything greater than a cm or so, I think.)
For those interested, see things like quadratic forms or metric spaces, etc.
Most people will be deterred once they actually find out what goes into basic general relativity physics - not because it's too difficult in principle, but because it's, well, boring. Riemannian geometry is what it's all about, and unless you have a vested interest in learning it, it's all a lot of tedious algebra.
WALOGIMBAT
Now this has me confused!
Without
Any
Loss
Of
Generality
It
May
Be
Assumed
That