No, the composition of two bundle projections needn't be a bundle projection.     
 It is already not true that the composition of two covering maps is a covering map.       
You can find a counterexample in Spanier's classic [*Algebraic Topology*][1], Chapter 2, §2, Example 8, page 69. 

However the composition   $g\circ f :X\to Z$   of the covering maps $f:X\to Y$ and $g:Y\to Z$
*is* a covering map as soon as $Z$ has a universal covering map (that is as soon as $Z$ is locally pathwise connected and semi-locally simply connected).    

**A misunderstanding ?** Spanier's counterexample (and there are others) seems to contradict   assertion (1) of your question.  Could you please supply an argument or a reference?






[1]:http://books.google.fr/books?id=h-wc3TnZMCcC&printsec=frontcover&dq=spanier+algebraic+topology&hl=fr&ei=_15lTp61C8Pj4QTU0vDHCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCsQ6AEwAA#v=onepage&q&f=false