Generally speaking, you can still file a lawsuit against a negligent driver even if you were doing something you weren’t supposed to be doing at the time of the accident. However, your recovery may be limited by the percentage of fault attributed to you for the accident.
While there is no statewide law requiring helmet use while cycling, (Ft. Worth requires helmets for riders under 18) the insurance company may argue that lack of a helmet contributed to accident injuries. The same goes for a lack of proper reflectors or riding on the wrong side of the street.