There is a statement, which says:
"Composites of homotopic maps are homotopic."
But there is something, what i don't understand:
1) if we got two homotopies:
$F:X \times I \rightarrow Y $ and $F': Y \times I \rightarrow Z$
How is it possible to composite them if $F$ goes to $Y$ and the domain of $F'$ is $Y \times I$ ? Should we use just $Y \times {1}$?
You don't simply compose the homotopies, you construct a new one.
Suppose $f, g : X \to Y$ are homotopic via $H : X \times I \to Y$, and that $h,k : Y \to Z$ are homotopic via $H' : Y \times I \to Z$. You need to construct a new homotopy, say $K : X \times I \to Z$, from $h \circ f$ to $k \circ g$.
There's a fairly natural way to do this: define $$K(x,t) = H'(H(x,t), t)$$ and check it has all the required properties.
I can't really tell what you're asking in your second question; I think you're using $f_1$ to mean too many things. If you clarify it I'll try to respond to that too.