According to Merriam dictionary, that has many usages, but those that I was interested in, and the examples are about are:
- as relative clause
- as conjunction to introduce a subordinate clause
It is similar to "ke" in Persian, however I think we may do an extra position in our sentences (which maybe is not grammatical in English)
In general, we can always find what "that" refers to. In the cases above it could be a thing, person (in relative clauses) or a sentence (in subordinate clauses)
So let's analyze the sentences,
The students who were absent, that I prefer not to mention their names, should do this practice...
Here that (in my intention) refers to a subordinate clause as it is in I prefer that you have a master degree , it could be
The students who were absent, I prefer that not to mention their name, should do this
But as in English we can say prefer to do or prefer not to do, then the correct sentence is:
- The students who were absent, I prefer not to mention their name....
if that points to the students then the correct sentence is
The student who were absent, whose names I prefer not to mention, should do this
In case 3 "that" was extra positioned, and the correct sentence could be:
- you said that you won't go there, ok ...
I feel we do extra position in Persian to emphasis something.
In sentence 2 and 4, that could refer to the clauses "a car suddenly stopped in front of us" or "he came"
Here, we said about something surprising (suddenly...) then "that" can make emphasis... or maybe it is like when to point the coincidence
I also know that can be used to express the result or purpose of something like (From google):
expressing a reason or cause. "he seemed pleased that I wanted to continue" expressing a result. "she was so tired that she couldn't think" expressing a purpose, hope, or intention. "we pray that the coming year may be a year of peace"
The case 6 is similar to expressing a result...
In all the other sentences, that refers to a sentence or clause. (And it is not much odd in English too)