For hyperbolic geometry, you can find the answers to your questions on [Wikipedia](http://en.wikipedia.org/wiki/Hyperbolic_triangle).  The interesting thing is that there is an absolute upper bound on the area of a hyperbolic triangle, even though lengths are unbounded.  

For spherical geometry, you can find the area of a spherical triangle [here](http://mathworld.wolfram.com/SphericalTriangle.html) and corresponding trigonometric formulas [here](http://mathworld.wolfram.com/SphericalTrigonometry.html).