To start with, do what is necessary to obtain a valid I.d. Get a birth certificate and if you are in the United States, a reissued Social Security card. Try to find a friend or family member who can accept a piece or two of mail for you.
As to survival in the elements, this will be a matter of where you are located. It seems you are in a cold climate area.
Tips for keeping warm include dressing in layers. Several layers of natural fibers, not synthetics. Natural breathes better and keeps a person from perspiring as much.
Newspapers. It may sound weird, but newspapers can shield against wind pushing through your clothing. Additionally, newspaper scrunches up which traps some pockets of warmth.
Try to eat somewhat late, as close to snooze time as possible. It can stimulate internal warmth which will help you doze off.
Stay in areas that are less windy. Some streets...especially nearer to water, seem to be windier, also certain intersections due to buildings being built and creating wind tunnels.
Waterproof footwear and clothing is the best way to go, you need to stay dry.
Always wear a hat and wear a scarf if you can or at least keep your shirt buttoned all the way up.
Look around, ask other homeless folk how they stay warm. You can still stay of on your own, but learn from them.
Sleep on the ground or a bench or wherever is available and try to get something between you and the ground. cardboard works well and you can place it b the trash bins when you wake up.
Sometimes a fire department will give you a blanket. Not always. If you are given one, hold on to it. I know one man who has an irritating tendency to throw the blanket out the following morning. I gave him a blanket once and the next day he had tossed it.
Another thing to try, get yourself admitted to a hospital. Once there, a social worker can usually assist in getting you the needed I.d. The difficulty with getting admitted to a hospital is of course needing to have a medical condition of some sort. I do know a guy who went to the emergency room claiming he as suicidal. It got him a hospital bed for three weeks after which he was directed to a shelter. Once in he shelter system he worked with the officials there and eventually was able to get his own place (government subsidy helps towards his rent).
I hope you can get yourself back indoors soon, before winter takes a hold.