You will not be able to sue your employer for providing a reason for termination you feel is false to the unemployment system. It is a "closed system" You file a claim, information is gathered from all parties, and a decision is made about your eligibility. In the case of someone who was fired, it would be necessary for the claimant to show that the employer did not have a valid work related reason to terminate the person in order for them to be approved.
If you do not agree with the decision that is made initially, you will be given the opportunity to appeal this decision. The reason that they (the employers) have provided are considered and the reason you give will be considered. The evidence presented by both parties has equal weight. The agency will work with the assumption that either or both the parties involved may not be being truthful. If they hear differing stories, hey will make their eligibility decision based upon which of the differing stories is "most believable." The other party always has the right to the agency appeals process. But it is not a civil court matter, it is an in agency matter. So you cannot sue for defamation or something like that based on what goes on in the unemployment system. Information gathered by the agency regarding claims and decisions made are not public records. They can be accessed only by the participating parties.
If you were injured at work, you'd need to be covered by workers comp. Unemployment insurance is not going to be an option until you are ready and able to return to the work world though you may want to go on and file the claim to establish it before you are actually able and available for work. You can apply, have a claim established that will be good for a year, and if applicable, you can begin the appeals process, but until you are able to work again, you cannot receive money from the unemployment claim, regardless of whether you were approved to receive benefits or not.
With years of listening to people tell why they were fired, I hear what you say and agree with quincy, I feel like a lot of information is missing here.