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, 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?