With the exception of "goeth" (which was standard English in the past: "go" with the old third person singular ending "-th", now employed to sound archaic), these words are informal spellings of words/phrases that represent how these words are actually pronounced. You only see them written this way in informal writing, but I think the pronunciations themselves are pretty widespread and are used in a number of dialects. All of the words you mentioned (other than "goeth") are used in my dialect, for example.
All of these words have their own etymologies.
According to the Oxford English Dictionary, apparently "gimme" started out as a Philly thing:
We wish somebody would compile a Philadelphia ‘idioticon’. We have many local oddities: ‘Gi'me’, for ‘give me’.
The American: a national journal, 1883
According to StoneyB, this is what happening with "you" in "gotcha":
[T]he vowel in unstressed you will usually be reduced to /ə/; and in rapid speech the dental stops /d/ and /t/ followed by palatal /j/ (orthographic ‹y›) will usually "assimilate" to an affricate: /dʒ/ (=‹j›) and /tʃ/ (=‹ch›).
The earliest examples are of "got cher". The earliest example spelled with an "a" given in the OED is from New York, 1912:
Another witness testifies to the words of the man whose pistol had done the work. ‘I got cha,’ this efficient agent remarked with something of the craftsman's pride in a task well performed.
As for "gotta", it looks like the "ta" for "to" is a bit earlier than "gotta" itself, since it's sometimes spelled that way in Middle English. According to the OED:
But while tó adverb, retaining its stress, came at last to be written too /tuː/, the preposition, being usually stressless, remained at to /tʊ/, /tə/, and in dialectal specimens is now often written ta, tae, teh, ti, tu (meaning /ta/, /te/, /tə/), some of which forms are occasional also in earlier writing.
The earliest attestation given in the OED of "gotta" that ends with "a" (the spelling "gotter"* is older) is from New York, 1881:
I gotta talk to a reporter
As for "gonna", the OED says this:
Representing a regional and colloquial pronunciation of going to, with reduction of the unstressed vowel and assimilatory loss of the initial consonant of the second element.
Their earliest example is this:
Now, Willie lad, I'm ganna gie
You twa or three directions.
Poems, chiefly in the Scottish dialect, 1806
* This adding of an r is known as rhoticity. In a previous answer of mine about what dialect Hagrid (from Harry Potter) uses, I quoted the following:
A clue to Hagrid’s regional background may come from the rhotocity implied by the post-vocalic ‘r’ in syllables where in the standard pronunciation variant the schwa should be present: ter, inter, tergether, etc. This rhotocity survived only in areas west of London, south of Birmingham and in Lancashire.