The first example sentence is strange. It seems a complicated way to
say "Here, prime rib is cut with a steak knife, as [it] is normally
done."
The second sentence can be regarded as an elision of "as IT is",
meaning "as it exists now".
It means that the item offered for sale (the omitted "it") will be
sold in the condition it is in, with no guarantees.
From AHD:
" Idiom as is (Informal)
Just the way it is, with no changes or modifications: "
The third example means the same as
(The shirt is old), as (the coat is[old]).
It moves the word "is" to the front of the second clause without
changing its meaning.
The word "as" in the examples is a conjunction connecting two clauses.
it means the relationship or action in the second clause is the same
as or similar to that in the first.
Dictionary definition of "as":
https://www.ahdictionary.com/word/search.html?q=as
(look under "conjunction".
In your examples of "as is", there is a conjunction "as", followed by
the verb "is" whicih is part of the second clause.
So, "as is" starts out as a coincidental occurrence of a conjunction
followed by a verb from a following clause. But in the middle example
of "as is" in the sale of used goods, it has become an idiom with its
own meaning.
--
Jack