Hate to bring the serious to the answer, but I’m with Jiim Klein:
- Pidgins are called that because of their origins, rather than their grammar, although they do tend to be remarkably similar.
- “Foreigner talk”, the way people dumb down language when talking to non-fluent speakers, are informally called pidgins, and indeed foreigner talk is a major origin of actual pidgins.
- Language games are typically not called pidgins.
- The recurring features of pidgins are things like dropping grammatical markers, using unmarked inflections, very simple syntactic structures.
Now, I’m not up on my memes, coz I’m old.
- Lolcat is a mix of foreigner talk, baby talk (which has overgeneralised inflections rather than unmarked inflections), and all-out whimsy; I find it hard to believe that any real pidgin would use the I of I can haz cheezeburger?, let alone the are of I are crying cuz I are out of focuss.
- Doge (meme) has a syntactic frame much too restricted to be a pidgin (many mis-subcategorisation, much exclamatory, such ludic), and a far more subtle sense of modifiers than any pidgin would bother with.
- I don’t know Doggo. If Doggo is not Doge, and is exemplified in How did the doggo language start? • r/OutOfTheLoop: doggo does a bork and u r doing me a frighten—then again, too much play on normal English syntax, and too much play with wrong inflections and derivations to be pidgin-like: a pidgin would just cut it down to Yu mekim mi frait.
You point out the use of gerunds for tense in Doggo as a pidgin: a pidgin is not going to know what a gerund is, because pidgins drop all the grammar they can. Tense in pidgins are separate words; the classic English-based pidgins use words like by-and-by (future), finish (perfective), been (past).
I mean, if people stuck on a plantation with no common language but what the masters barked at them spontaneously started speaking in Doggo, then yeah, Doggo would be a pidgin. But what I’m finding doesn’t look like a pidgin. What I’m finding is comically inverted English, rather than radically stripped down English.