According to [Merriam][1] 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:

> 1) 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:

> 3) you said **that** you won't go there, ok ...

I feel we do extra position in Persian to emphasis something.

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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 

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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)


  [1]: http://www.merriam-webster.com/dictionary/that