To address the first question, I suggest you have a look at *canonical correlation analysis* and to a more recent dimension reduction technique called *sliced inverse regression*.
On the latter, see the initial paper by Ker Chau Li : Sliced inverse regression for dimension reduction (with discussion). Journal
of the American Statistical Association, 86(414):316–327, 1991.

It is freely available on the Internet. The version with the (interesting) comments you might have to buy thought.

Some important parameters for the choice of a method in you situation are :
 - dimensionality of the input (n=3, n=15 and n=50 are very different problems);
 - time needed to get one evaluation (0.1 s, 5 min and 5 hours are also very different problems);
 - assumptions that you can make about your model : is it linear ? is it monotonous ?

Also you mention a possible multivariate output. If you have a few of them that represent completely different things, just do several independent sensitivity analysis.

If they are higly correlated or functional then it also change the problematic a lot.

You should make all this points clear before going for a given methodology.