If Jews claim everything is predestined, and that we can do nothing to change it, why do we look before we cross the road? I'm a Jewish man myself and personally never look before crossing and have yet to be hit. It's not like we are going to die before our time.
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Although you are correct in stating that everything is predestined and that looking both ways before crossing the street really does not protect one from death, one must nonetheless adopt the practice as a chesed (kindness) to those who are fated to die as they cross the road. Although it is a fiction, we should pretend that looking both ways does protect one from death. When someone looks both ways, even if he knows on an intellectual level that it is of no consequence, he does get the illusion of protection, which is comforting.
First, G-d's omnipotence does not preclude free will, as the statement you bring clearly states. Omniscience does not mean that everything is decided already, only that G-d knows what you will choose to do, even if you don't know yet.
But it's true that certain things are decided in advance , so the question still stands, albeit on slightly altered legs.
But the statement that "we can do nothing to change it" is categorically untrue. Judaism believes in the efficacy of prayer, and that G-d empowers us to change even that which has been decreed above through prayer, repentance, and good deeds. Some of the most important prayers we say on Rosh Hashana and Yom Kippur spell this out quite clearly.
Your more direct question is "if the method and time of a person's death is already decreed, why not do dangerous things?" Well, the Talmud tells us that "HaSatan mekatreig b'shaas sakana". ("The Adversary prosecutes in a time of danger.")
That means, when a person is in a dangerous situation, the accusing angel (or 'the Satan') goes before G-d and basically says, "Look this guy deserves to die for his sins, and now is the perfect opportunity for me to do it. May I?)
Therefore, we avoid dangerous situations to avoid giving the Accuser an extra opportunity to prosecute us.
In Jewish thought it is by no means agreed upon that "everything is predestined." That everything is foreseen, perhaps, and that does create a certain paradox (see the Rambam in Hilchos Teshuva 5:5); but that everything is predestined, certainly not.
According to Maimonides, man has complete freewill when it comes to the realm of action. In other words, according to him, you have freewill over any choice you make. Maimonides is clear about this in Chapter 8 of his שמנה פרקים, his introduction to Pirkei Avos.
If you walk into the street without looking and get hit by a car, the Rambam would say you are completely at fault.
Mesilas Yesarim, Chapter 10 writes:
Translation credits: http://www.rabbisilber.com/wp-content/uploads/2014/10/MESILAT_YESHARIM.pdf