For IUPAC nomenclature, does the number of multiple bonds take precedence over the chain length?
According to the new recommendations of IUPAC nomenclature rules, when choosing the senior parent structure, you have to first consider the senior parent structure, which has the maximum number of substituents corresponding to the principal characteristic group or senior parent hydride in accord with the seniority of classes. Then, if there is a choice, choose the principal chain, which has the greater number of skeletal atoms. Next in order is the chain, which has the greater number of multiple bonds, and then has the greater number of double bonds (This is according to Blue Book in 2013, which was displayed in 5 steps you have mentioned from Wikipedia).
Note that the new IUPAC recommendations (Blue Book in 2013) on order of chain length and number of multiple bonds have been changed (You can find more details here and here):

In acyclic parent structures the order of seniority between unsaturation and length of chain given in earlier recommendations is reversed. Thus, the first criterion to be considered in choosing a preferred parent acyclic chain is the length of the chain; unsaturation is now the second.

For example consider following structures:

In all first three structures, the longest chain has chosen over the multiple bonds (the old names are in orange). In the typical case, the principle group is $\ce{-CHO}$ and hence hence to hexanal is the principal structure which has $\ce{-OH}$ substituent and longest chain. Thus, the name is 2-ethynyl-5-hydroxyhexanal.