help with inusrance

No, don't confuse public & property owners liability as they work in different ways.

Home insurance includes Property Owners Liability which covers you for any third party injury occurred due to your negligence on YOUR land or inside YOUR home - not anywhere you happen to be. Good home insurance will extend this to Personal Liability which will cover you for third party damage caused by you - you'd need to consult the wording though to confirm this.

Public Liability is usually applicable to businesses, contractors, window cleaners etc and covers damages to third parties if negligence can be shown - window cleaner breaking a window or somebody tripping up in a shop if the carpet has been badly maintained etc.

Specialist cycle only insurance will usually give you an option for Personal Accident, which covers you for injuries to yourself. This is often based on your age, and will exclude injuries sustained while racing and time trialling though. It's not something you would usually get thrown in on a home insurance, and given in the UK we have the NHS and statutory sick pay, it's not cost effective to take out separate cover for most people.
Your Home Insurance, as long as the bike is specified, will usually pay for the costs of repairing or replacing your bike, subject to two estimates (you have to get at least two to prove you are mitigating your costs). In my experience, insurers usually have their own supplier and will offer you a choice of bikes to the value of your old bike if it's written off, less any excess applicable on your policy. Any damage to a third party would come under the Public Liability section of your home insurance, and the onus would be on the Third Party to provide proof the damage you caused was your fault, and they would have to substantiate their costs too.

Depends. It may be difficult via an "open market" insurer such as Direct Line et al as their wordings are generally so full of warranties and clauses that they'll find a way to wriggle out of it.

On the other hand, a good broker who also rides (like me) can find you a policy that includes cover for accidental damage, theft or loss anywhere in the world and provides personal liability for any damage you may cause to third party property anywhere in the world. Generally, this cover combined with your home contents and jewellery/art etc works out cheaper than a policy from somebody like ETA who just insure the bikes.

As for injury to yourself, I'd have to look further into this.