Simply leaving off the G on words like happening, or writing "an" for "and" is not informal or colloquial - in my opinion, it's suitable ONLY if you are writing dialogue and want to show how the person sounds.
As for "ye" I have never come across that in any sort of modern writing or speech.
"Gonna" is the only only I would find acceptable as colloquial, and NOT as informal. When English learners write things like "im gonna have a test in E next wk n i need ur help" it makes me want to scream. In a text message, I don't object to "gonna" and I can live with it in an e-mail between friends. Not in a post here, where people are supposed to be learning proper English.
So in short, No, No, No.
As for the abbreviation, it's common to spell it out the first time with the abbreviation in parentheses after it. The National Solar Observatory (NSO). The National Science Foundation (NSF). All future references to either one can be simply NSO or NSF, as applicable.
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