Did the store owner assume responsibility when they stated have the bill sent to them?

The fact that the store owner instructed you to go to a doctor and send them the bill SUGGESTS that the store knew it was responsible, however, by itself, it does not win your case against them. They may explain their offer by saying the store owner was only trying to help you, whether or not he was liable for the injuries, and a jury (if the case goes to trial) might believe that. You need to hire a lawyer and, if necessary, sue the store owner for your injuries. The fact that he and his insurer SAY they are not liable does not mean they aren't.

No, he didn't. Get a lawyer. You may be entitled to more, and you may be entitled to nothing.

If there is fault you have a claim. The insurance company thinks there was no fault. If you don't prove fault you don't have a claim. You get nothing because you fell. You must prove fault.

Consult an personal injury attorney. Generally stores are responsible for split and falls. They have to show that you were negligent. From the facts you stated it does not sound like you were negligent. A personal injury attorney will take the case on a contingency fee basis if you have a case. That means you would not have to pay the attorney until you settle or win.

Depending on the size of the medical bill it probably would not be a big enough claim for an attorney to take it. You could take the store owner to small claims court, but they would be able to bring an insurance adjuster to represent them and fight it and you would have to prove the store was liable, i.e. failed to salt the ice when they knew it was icy.

You should talk to the insurance adjuster again. It is typical that in commercial liability policies there is medical pay coverage that will pay an injured person's medical treatment bills up to the limit of coverage and without regard to whether the insured store was at fault/negligent. I do not believe the insurance company and/or agency is being "straight" with you. If the store owner is a person of their word, they may agree to request the insurer pay your bills out of the med pay coverage and/or show you a copy of their declaration sheet for their insurance which would reflect whether or not med pay coverage exists, and if it does, in what amount. The owners promise to pay your bills may serve as a "contract" that you can enforce - or maybe you could assign your rights to such enforcement to the medical care provider and let them file the claim/suit in your place? As I've said in many other posts, you only have a remote chance of succeeding on a premises liability claim in Michigan for removal of snow/ice due to the "open & obvious" doctrine. However, if the store owner increased a hazard vs. a natural accumulation, you may have a basis for such a liability claim. If there was such a claim it could include any and all medical expenses, as well as pain & suffering. You would need to consult local counsel to review your facts re: whether such a claim was even viable with your facts.

You have the right to formally open a claim with the insurance carrier, or even file a lawsuit to have them cover medical expenses, as well as pain and suffering. I would obtain services of an attorney for assistance in filing a suit against the store.

Yes, when they told you to go to the hospital and send them the bill, they committed to pay for it....if their own insurance company refuses to pay for it, then they have to pay it.

You need to file a claim with the grocery store's insurance company. If they won't pay immediately, you should file a lawsuit against the store owner.

You can sue the store owner in small claims court for the medical expense. What made him liable is not only his undertaking to pay the bill, but the fact that you relied upon his promise when you incurred the bills for medical treatment.

In my opinion, they did not. They made a one-sided promise to do something in the future and then changed their mind. That is not a legally enforceable obligation.

No, just saying have the bills sent to us is not an admission of fault. It's a store's way of getting rid of you at the moment and giving anyone in earshot the notion this store takes care of it's customers when in fact the opposite is true.