>> Any help would be fine

All but one of your examples were correct.  The following example was not quite correct:

"Jerry failed his test again, he has lately come across as a lazy student."

The following definition is not quite correct:

"To come across" describes the way someone is or how someone acts in particular situations.

If you look at the bottom-right box of your photograph, you will see an important word: "appearance".  Therefore, the following definition may be better:

"To come across" describes the way someone <em>appears to be</em> in particular situations.

Let me give you some examples of this use of the phrase "to come across":

"He is a brilliant professor, yet outside university he comes across as a kindly old man."

"He usually comes across as a kindly old man when in truth he is a brilliant professor."

"She studies extremely hard in her bedroom, but in class she comes across as a lazy student."

"In class she comes across as a lazy student when in fact she studies extremely hard in her bedroom."

This use of the phrase "to come across" suggests that a person's <em>appearance</em> is different from his or her <em>real</em> self--at least in a particular situation.

Let's look again at your fourth example:

"Jerry failed his test again, he has lately come across as a lazy student."

Because Jerry is <em>actually</em> failing, his laziness may be a real cause and not an "appearance" of laziness.  A better example would be:

"Jerry scores well in all his tests, yet lately he comes across as a lazy student."

I hope this helps.