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.
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:
The Sages in all places have ordered that a man be especially attentive to his well-being and not put himself in danger even if he is righteous and a doer of good deeds, that they have said (Kethuvoth 30a), "All is in the hands of Heaven except chills and fever," and that the Torah states (Deuteronomy 4:15), "Be very watchful of your selves" - all of which indicates that a person is not to extend trust in G-d to this area, even (as our Sages state further) when a mitzvah is to be performed.
Know that there is fear and there is fear. There is appropriate fear and there is foolish fear. There is confidence and there is recklessness. The Lord blessed be He, has invested man with sound intelligence and judgment so that he may follow the right path and protect himself from the instruments of injury that have been created to punish evildoers. One who allows himself not to be guided by wisdom and exposes himself to dangers is displaying not trust, but recklessness; and he is a sinner in that he flouts the will of the Creator, blessed be His Name, who desires that a man protect himself. Aside from the fact that because of his carelessness he lays himself open to the danger inherent in the threatening object, he openly calls punishment down upon himself because of the sin that he commits thereby, so that his hurt results from the sin itself.
The type of fear and self-protection which is appropriate is that which grows out of the workings of wisdom and intelligence. It is the type about which it is said (Proverbs 22:3), "The wise man sees evil and hides, but the fools pass on and are punished."
"Foolish fear" is a person's desiring to multiply protection upon protection and fear upon fear, so that he makes a protection for his protection and neglects Torah and Divine service. The criterion by which to distinguish between the two fears is that implied in the statement of our Sages of blessed memory (Pesachim 8b), "Where there is a likelihood of danger, it is different." That is, where there is a recognized possibility of injury, one must be heedful, but where there is no apparent danger, one should not be afraid.
Translation credits: http://www.rabbisilber.com/wp-content/uploads/2014/10/MESILAT_YESHARIM.pdf
I learned from one of my teachers that sometimes the Heavenly decree as to what will happen to a person is dependent on whether he has merits or not. For example, there may be a Heavenly decree saying "If this person has enough merit he will succeed in his business, if not he will not. Additionally, when one is in a situation of imminent danger more merit is needed to save him from a disastrous result. Therefore, one should look before crossing the street so in case he is in a situation where without the danger he might merit continued life and health, but he does not have enough merit to be worthy of being saved from the imminent danger, he should not get himself into a dangerous situation.