Well, it's correct in the sense that "oi fink" is a correct pronunciation of "I think" because that's what some native speakers say.
It's not something for a learner to emulate unless they want to live in an area where people talk like that, though -- for that, a more "general" accent is best, rather than a regional one.
No, Hau-sch-tier is never correct in german, even in Dialects were "st" is almost always pronounced "scht", like in my own - bavarian!
It is because two separate words have been combined to a single one but in German the pronounciation never "melts" together like a liaison in French for example.
Pronunciation is wrong, even for South Germany. The engine for pronunciation does not recognize that this word is composed of two words. In South Germany you can pronounce st as you hear it, but not if the s and t belong to different words. There is only one region, swabonia, where s may be always pronounced sch, but that is not swabonian dialect... Husband of poster lived there for >10 years.
Etymologically speaking, tier and deer are closely related and come from the same word. (English being a Germanic language 1000+ years ago). Scandinavian languages have "djur/dyr", which also means animal (in general) like German tier. For some reason, in English the word became more specific to apply to only one specific type of animal, while in other Germanic languages it became more general to include all animals.