Term for words like Snowmageddon, Nipplegate and even cheeseburger?

I don't know if this is quite the term you're looking for, but you could refer to such clever wordplays as abstractions.

Etymonline says, under its entry for -gate:

-gate: suffix attached to any word to indicate "scandal involving," 1973, abstracted from Watergate...

I wondered if that was a one-time mention of the term abstraction, or if it was used at other places on the site, and I found similar phrasing under entries such as -aholic, -rama, -fest, -athon, slimnastics and yes, even cheeseburger:

-oholic: word-forming element abstracted from alcoholic; first in sugarholic (1965), foodoholic (sic., 1965); later in workaholic (1968), golfaholic (1971), chocoholic (1971), and shopaholic (1984).

-rama: noun suffix meaning "sight, view, spectacular display or instance of," 1824, abstracted from panorama (q.v.), ultimately from Greek horama "sight, spectacle, that which is seen."

-athon: word-forming element denoting prolonged activity and usually some measure of endurance, abstracted from marathon. E.g. walkathon (1931), skatathon (1933); talkathon (1948); telethon (1949).

slimnastics (n.) 1967 (with an isolated use from 1959), from slim (adj.) + ending abstracted from gymnastics.

cheeseburger (n.) 1938, American English, from cheese (n.1) + ending abstracted from hamburger.

This won't give you a complete list, you can peruse more abstractions from the website here.

This sort of portmanteau is closely related to the phrasal templates called snowclones:

Snowclone is a neologism for a type of cliché and phrasal template originally defined as “a multi-use, customizable, instantly recognizable, time-worn, quoted or misquoted phrase or sentence that can be used in an entirely open array of different variants.”

An example of a snowclone is the phrase “grey is the new black,” which gave rise to the template “X is the new Y.” X and Y may be replaced with different words or phrases—for example, “comedy is the new rock ’n’ roll.”

You could probably call such a word a snowclone and be understood, although if you want to emphasize that it's a single word, you could call it a “snowclone word” or “snowclone portmanteau.”

I found a list of -gates on Wikipedia (with a bonus sketch on YouTube).

That Wikipedia article also references a short text, Yet more on -gate words. It doesn't mention a term, but I suppose "-gate words" works for that particular class.

If I were to try to come up with a term for how the suffix acts, I'd perhaps go with productive suffix. Which I then googled and found on Wiktionary. That list includes "-punk" (e.g. "steampunk"), but not any of your examples.

Googling further from my findings, I came across a Wiktionary entry for burger which calls the "-burger" suffix a back-formation (list). They're referrering to the suffix, but a comment considers "cheeseburger" a back-formation also. A book calls it backformation without the hyphen.

This text on word formation uses the term blending for "cheeseburger" and other words that are not joined along morpheme edges (assuming this one is "cheese" + "hamburger"), and it says the word "burger" is formed by clipping.

Long story short, I think I'd go with back-formation :)