None of those fancy words are correct in the context you used them.
Should it be
None of those fancy words are correct in the context in which you used them.
Or :
None of those fancy words are correct in the context that you used them.
With regard to your first point: yes, the sentence is grammatically acceptable without preceding the clause "you used them" by 'that' or 'which'.
You are using what is called a 'zero relative pronoun', which—although uncommon in other Germanic languages—does exist in English.It is also sometimes referred to as a 'zero clause' or a 'contact clause'.
The 'that' or 'which' is implied here, and can be tacit, i.e. not explicitly written or spoken.
The next issue depends on whether "you used them" is a restrictive relative clause or a non-restrictive relative clause.
A restrictive relative clause can be preceded by either 'that' or 'which'; a non-restrictive relative clause should never be introduced by 'that'.
'Restrictive' means a clause which contains essential information about the object-noun (which in this case is "context") while a non-restrictive clause contains merely extra information. To decide which is which, ask yourself if the sentence makes the desired sense without the clause. If it does, the clause is non-restrictive.
Note also that a non-restrictive relative clause usually requires a comma beforehand.
Examples of a restrictive relative clause:
"After test-driving two cars, Mary bought the one which had power steering." (correct)
"After test-driving two cars, Mary bought the one that had power steering." (correct)
Both sentences are correct because the clause "had power steering" is essential in defining which car Mary bought.
Examples of a non-restrictive relative clause:
"Yesterday Mary bought a great new car, which made her happy." (correct)
"Yesterday Mary bought a great new car, that made her happy." (incorrect)
The second example is incorrect because a non-restrictive relative clause should not be introduced by 'that'.
Having said all that, 'which' usually sounds better than 'that' for a restrictive clause; your second sentence sounds better than your third, even though strictly speaking both are correct if "you used them" is a restrictive clause.
All of those sentences are correct. As it's a restrictive clause "that" is more commonly used; additionally "in which" sounds clunky. Therefore it would be better to use "that", but all are correct.
The first sentence is an example of reduced relative clause. These are perfectly acceptable, so long as they don't give rise to ambiguity; which yours doesn't.