The combination of the two halves is not a general closure theorem. An fpt-reduction is only required to run where its image fits in the word length, and the image here has a parameter-sized table, so on a word much shorter than that table the decision program cannot run the algorithm for the integer programs on the image; the proof of $fpt_byClients$ handles those words, whose number of schedules is bounded by a function of $n$ alone, by enumeration. On every other word it writes the integer program and runs the algorithm for the integer programs on it at a word length chosen for that program.

Hardness is stated at treewidth at most $6$ rather than for each $\tau \ge 6$ separately. The class of instances of treewidth at most $6$ is contained in that of treewidth at most $\tau$ for every larger $\tau$, so the statement at $6$ gives every other one; the source's construction forcing the treewidth to be exactly $\tau$ serves the version of the claim in which the parameter is pinned rather than bounded.

The two tractability claims are claims about one program each, on the word encoding of an instance, with the parameter a function of the word: the number of days plus the treewidth of the overall conflict graph of the decoded instance, and the number of clients. The number of days and the number of clients are entries of the word; the treewidth is a structural property of what the word encodes, which is a function of the word because a word encodes at most one instance.

A word of the domain is an instance block followed by the single entry of the fairness parameter, so the instance is decoded from the word without its last entry ($x.dropLast$); the parameter itself is the last entry. Decoding the whole word would read no instance at all, since the block has even length, and would make both problems trivial.