The parameter is read off the input word rather than carried beside it. This is what lets one program serve every parameter: a program that had to be told $k$ from outside would be a family of programs, one per parameter, free to hide unbounded advice in its literals. The quantifier order says the same thing — the program and the constant come before the instance, the parameter and the word length.

$Fits$ is the fitting condition, an explicit inequality against $2^w$ rather than a statement about logarithms. It says of each entry $v$ of a word that $c(|x|+v+1) \le 2^w$: every entry is a genuine word, with room left for the constant multiple of the length that bounds the addresses and the step count. Entries are not bounded by the length here — a due date or a weight may be any number at all — so quantifying over them is what makes a claim about them honest rather than a silent assumption that they are small.

A reduction's output has to fit as well, since a machine at word length $w$ reduces what it writes modulo $2^w$. Like every fitting condition this restricts the inputs rather than appearing as a hypothesis: as a hypothesis it would be empty, because no word length accommodates every encoding of a fixed instance at once.

The bound is elementary, $c * g k * (x.length + 1) ^ c$, with the $+ 1$ making it meaningful on the empty word. The same constant serves as the multiple and as the exponent, which costs nothing — raising either is raising both — and keeps one number to quantify. $g$ is an arbitrary function of the parameter: it bounds a fixed program's running time rather than defining it, so no computability requirement on $g$ is intended.

The polynomial factor is what the definition of fixed-parameter tractability asks for, $f(k) \cdot n^{O(1)}$, and it is what a reduction needs here: the construction of this submission emits an instance polynomially larger than the one it reads, so no reduction computing it can run in time linear in its input.