1) Sorry, but weather has no bearing on this! You were in an accident that was your fault. An accident can NOT be Mother Nature's fault. Consequently, weather has zero bearing on this since you'll be found to be 100% at fault since it was a single vehicle accident. After all, insurance companies can't sue Mother Nature in court to get money back for all the money they pay out for accidents she causes.

2) Most states require that any accident above a certain dollar value MUST be reported to the police. You should look into this in your state. I suspect you'll have to file an accident report for this since it's $5000. This is a separate issue than whether it'll affect your rates in the future. But the answer is yes, other insurers will be able to see it. They don't use a DMV database like they do for traffic tickets, nor are accidents stored in a DMV database unless there's an associated traffic ticket issued. There's a separate database insurers use to track claims that are submitted, and this would go there.
3) You'll have to talk to your insurer about the rate hike. That's much to broad a question for anyone other than them to answer!
Best of luck, and sorry to hear about your accident.