Agreed that 안 편하다 not comfortable and 불편하다 uncomfortable are synonymous.
But I reckon the grammar point here is about the placement of the adverb 안 Not when used with -하다 verbs.
DLG wants to point out that when used with a descriptive -하다 verb (i.e. an adjective), 안 cannot split the verb and goes in front of -하다 as in the case of most -하다 action verbs, i.e.
Not comfortable = 안 편하다, not 편 안 하다.
In fact, 편 안 하다 may be misinterpreted as 편안 하다 which means exactly the opposite, be comfortable (편안, comfort).
자다 = sleep
차다 = kick, attach, wear ...
But to be absolutely certain there is no confusion, you can use the full idiom 잠자다, to sleep.
차다 and 차 on the other hand are tricky words as they do carry various meanings. Best to consider the whole sentence than trying to dissect individual words.
In English, the "definite article" (the) is used to indicate a specific object [unlike the indefinite article (a or an) which means "one or any".].
When translated into Korean, the "definite/specific" function of "the" tends to be reflected in modifiers such as 이 (this); 그 (that there); 저 (that) .
Your sentence would look something like:
그 허리띠를 착용하는 것은 안 편해요 = Wearing the belt (there) is not comfortable.
V-는•것 can only be attached to verbs. So in this case, we can safely assume "차" comes from the verb 차다.
But I agree "차" as a noun can mean car(s) or tea(s).
I came across a sentence recently "일본차는 매우 강하다". I am still not sure whether this means "Japanese teas are very strong." or "Japanese cars are very powerful."