What is (1) the meaning of the names and (2) the structure of the “poems” found in Selichos particularly the Pizmon and the Shalmonis. (3) I note that the number of phrases in the first verse of the Pizmon is less than in subsequent verses. Why is that?
All the slichot poems are called "slichot," but they can also have different names, based on their form, their subject matter, or their location in the davening.
This answer is specific to Ashkenazi slichot.
For Pizmon, the Lubavitcher Rebbe zt"l (in his notes at the end of the Chabad Selichos) summarizes the following explanations:
For Shalmonis, the Rebbe quotes two explanations, although without providing sources:
Given that the first verse (or part of it) of a pizmon is meant to be used as a refrain between stanzas, then it makes sense that it would be shorter than them, since the stanza would carry part of the melody and the refrain would carry the rest.