If your doctor accepts workers compensation all treatment relating to the work related injury only should be billed to the insurance carrier. The employer, if insured is not responsible to pay anything. If you fail to pursue your claim, you could be held responsible
If you are claiming unemployment you are holding yourself out as ready, willing and able to work. You can receive retirement benefits under Social Security and work, though there can be certain limitations on earnings. If you are receiving Disability benefits that means you are unable to perform any substantial gainful activity or perform any work currently available.