Is refraining from speaking with one's sister on the phone – while suffice with sending congratulatory messages on Eid – considered a severance of kinship ties, out of fear of falling into disobedience to Allah due to her unlawful actions?

Maintaining kinship ties refers back to custom. If it is customary that sending messages fulfills the connection, then the person is maintaining kinship ties. Your sister is disobedient by her display of adornment (tabarruj) and her interaction with non-mahram men in this state, as the hijab is an obligation. Allowing non-mahram men to see her in a state of tabarruj is a clear sin, and you are right in your protective concern (gheerah) and your denouncement of her. Your being younger than her in age is not a Shari'ah impediment to your denouncement of her. We advise you to be patient with her, treat her gently, and teach her matters of her religion with kindness and gentleness. Contact her and explain to her that you are speaking with her out of mercy for her. If she persists, you may forsake her (hajr) if you hope that the forsaking will deter her, for forsaking is resorted to based on benefit.