"She is making every effort not to cry."
Translation:היא עושָה את כל המאמצים לא לבכות.
That's just how the expression is. Note that it's slightly different to the English equivalent ("all of the efforts" instead of "every effort").
Also, i think the every changes it in english but hebrew doesnt have that word
It's just like with כל בית and כל הבתים (every house vs. all the houses). So כל מאמץ is "every effort" and כל המאמצים is "all the efforts".
The "correct" sentence out of three is
היא עושה את כל המאמצים שלא לבכות
Can שלא be used here instead of לא? Doesn't sound grammatical to me, since it is followed by an infinitive.
It's correct. The -ש can be added when the infinitive is negated, i.e. before לא.
The common way people use often not to cry is to activate sympathic vegetative nerves. Say to keep leg's or neck's muscles strengthed, stiffed for many days. Of course they pay it by chronic bad mood...