Does divorce occur if it is not uttered, but rather by filling out a divorce certificate and checking the box "second divorce" when a previous divorce has occurred, and sending a picture of the paper to the wife with the phrase "don't bother me, what was between us is over," in addition to relatives answering "yes" when asked about the occurrence of divorce?

It appears that merely placing a mark on the divorce paper does not constitute a divorce, as long as you did not utter the words of divorce or write them down. Signing your name is not one of the forms of divorce, nor is it an allusion (kinayah) or explicit writing. Your message to your wife is an indirect allusion, so as long as you did not intend divorce by it, it does not count as a divorce. However, your answering "yes" to those who asked you, "Have you divorced your wife?" does constitute a divorce, unless you intended to lie, in which case there is a difference of opinion among scholars regarding the validity of the divorce religiously (diyanatan).