> Can someone tell me some universal rules that apply > all the time no matter what type of geometry?
Euclidean, hyperbolic and elliptic geometry have quite a lot in common. For example, the Euclidean criteria for congruent triangles also apply in the other two geometries, and from those you can prove many other things.
To give a more historical answer, Euclid I.1-15 apply to all three geometries. Euclid I.16 and various later propositions fail in the elliptic plane. Euclid I.29 and various later propositions fail in the hyperbolic plane.
> My friend Brian asks: Is there a specific ratio > between hyperbolic sines and euclidean sines?
Are you willing to use complex numbers? If so, then sinh(iz) = i.sin(z), cosh(iz) = cos(z).