And now for my extreme point of view:

All languages change over time; the vocabulary, the grammar and pronunciations change over time.

Gonna, wanna, gotta, and others are spellings that simply represent the sounds, however, they are what I would call "lazy speech" (even though many people do it and I probably do it as well, although I try not to).

Those words are never acceptable in written form and never in formal writing, unless they are beng used to represent someone's speech.

I think we should take the time (approximately a fraction of a second) to pronounce the words properly, i.e., going to, want to and got to.

That's one way that I try to respect my language.

Good question Mikkel...

You have identified the salient point straightaway, you don't consciously notice when you say them. Neither do I. I must use them several hundreds of times a week, probably even thousands of times, but I don't consciously do so. 

They're not contractions. 'They're' is a contraction. They're approximations of how it sounds when native speakers, speaking naturally and reasonably quickly, say 'want to', 'going to' and 'got to.' Everyone does it.

It's definitely a relaxed form of speech, and native speakers usually do this (but, as mentioned, we almost never write that way).  If you're speaking like this because you are truly relaxed with English, then that's fine.

However, learners tend to make a few mistakes with these, and this is where things start going wrong. For example:

- they use "wanna", "gotta" and so on because they think it's a "better" or "authentic" way of speaking and writing.

- they assume these are real words, and therefore come up with impossible phrases such as "she wanna" or "I wanna to..."

- they either pronounce the expressions wrong (ie. the "a" at the end gets pronounced exactly as "a" and not a neutral schwa) or they never learnt to pronounce the "ng" in "going to", for example.

This is where I pull students up and make them repeat their sentences. Usually, native English speakers have no problem repeating their sentences in standard English.

All native speakers use reduced form pronunciation to some extent. It varies from person to person and even an individual will unconsciously use more or less reduction depending on social setting.

A few native speakers might say "djeet?" (Did you eat?) or "Meyelp?" (May I help?).

Most (but not all) speakers use the reduced forms you listed above.

However I think almost everyone pronounces the auxiliary "have" as a schwa (eg. could have = coulda), since that eliminates not one but TWO pesky consonants.

"Wanna", "gotta", "gonna" have somehow weasled their way into written English (yuck!), as have "kinda", "sorta" and "gimme". Other reduced forms that are seldom written include "woulda" (would have), "yuusta" (used to) and "hafta" (have to).