Perhaps the best answer is "yes and no". If your auto insurance company denied the claim of another party with whom you had a collision, it was either because you did not have liability coverage to protect you from the claim, or because your insurer, after investigating, did not believe that you were At Fault.
If the other party had liability insurance, the insurer can deny your claim for the same reasons.
If this happens, the other party is generally free to sue you, in which event you need to turn the suit papers over to the insurer so that it can defend you (if you had liability insurance). If you did not have liability coverage, you will have to defend the suit yourself. In all events, you have to act promptly, or else you may lose by default.
When you (or the insurance company) respond to the lawsuit, if you have a claim against the other party arising from the same occurrence, it must be asserted as a "compulsory counterclaim". If it is not, you will lose the right to assert it as a separate suit. If the other party had liability insurance that covers your counterclaim, it will defend the counterclaim. The ultimate result will either be that the claim(s) are settled or resolved by the court after a trial on the merits.
No, they will file a claim with their insurance company and their company will talk to your company. Assuming you swapped insurance details.
Then it's time for a lawsuit.
Let your insurance company/agent handle the claim--they will collect from the other insurance if there is a valid claim. You'll need a police report.
Read your policy
Same rules apply as if it was your own car. Make sure you get the insurance information from the person that was at fault and immediately call that insurance company and file your claim. The owner of the car will probably have to verify that you had permission to drive the car. If the other persons insurance denies the claim then the owner of the car will have to make a claim under their policy.
If they denied that claim for cause, such as fraud, then there is nothing you can do. If they did not have cause, you can try contacting the department of insurance in your state. That is of course, as long as you had comphrensive coverage on the policy. If you did not have coverage, they do not have to pay for anything.
A claim is a liability on part of the insurance company. If a customer makes a claim it means that the insurance company has to pay the customer for the amount is eligible to claim and hence it is a expenditure on the balance sheets of the insurance company.
Contact the insurance commissioner for that state and file complaint
The only step you have left is to sue the driver of the other vehicle. Keep in mind, it is your responsibility to prove that the other driver was at fault.
In the US, at least, the answer is yes. You can sue just about anybody for just about anything.A good resource for you would be your own insurance agent. Ask him/her about how to get an insurance company to respond to the claim.
That's not very likely. The insurance company does not file your claim, they accept your claim notice from you. You have to file your claim with the company, not the other way around.
Yes. That is part of your claim against them. However, if you filed with your insurance company, you gave up your right to pursue them for damages. Generally speaking, your insurance company will pursue the other party's insurance company and if the other company pays, that payme usually includes your deductible.