Typically, your renter's policy will not cover your hotel due to your air conditioner being out. A broken AC is not considered the loss of use of the dwelling (home) — they typically only pay for other accommodations when the entire dwelling is uninhabitable. I recommend you check out our guide to what renters insurance covers. Every company is different so it doesn't hurt to call your insurance company and ask a claims adjuster how your medical issues may affect your policy. If you have any questions, don't hesitate to ask.
The new insurance company is likely to find out about the accident, but that will come from your driving reports when you buy your policy. The majority of insurance companies request a motor vehicle report (MVR) and a comprehensive loss underwriting exchange (CLUE) report to verify the accuracy of a new customer's driving record.
For the purpose of getting your belongings replaced as quickly as possible, I would definitely recommend contacting your renters insurance provider immediately. While your neighbor will likely be considered liable for the water leak, your insurance company should be able to act quickly to replace your items and even provide a temporary place to stay while your apartment is being repaired.