How do we know which lighter elements fuse to form heavier elements? Formation of lighter elements is fairly straightforward:  $^4$He + $^4$He + $^4$He = $^{12}$C.  However for each heavier element (heavier than Fe in particular), is there exactly one combination of (two lighter) elements that can be fused?  Is it possible that any combination of atomic weights can be combined to form a heavier element; i.e., can any combination of sufficient atomic masses combine to form a given heavy element?
 A: You are, in fact, correct : there's more than one way to combine elements to form other heavy elements.
What we can predict to some extent is the relative probability of different combination (different fusion reactions).  We also have knowledge of how relatively stable the different nuclei are.  Some reactions are much less likely (by orders of magnitude) than others, and some of the fusion products are themselves very unstable.
The subject area, in case you need to know, is called Stellar Nucleosynthesis.  There is a link on that page to an old paper by Burbridge, Fowler and Hoyle which gives a good flavor of how this is worked out.  Things have advanced somewhat since that paper, but it's an interesting read (it's a long paper, almost a book - 100 pages).
It's not really practical to describe all the possible reactions in a star so we tend to just concentrate on the main ones.
We can check our predictions using evidence from various sources on the cosmic and stellar abundance of various elements.  So we get to know that our theoretical calculations, while we can't be necessarily exact, are in the right ballpark.
