Legal Question in Wills and Trusts in Georgia

Your question does not really make sense. The state just doesn't pick a person and decide that they are "next of kin." You admit that you are a cousin so you are probably within the degrees of relationship allowed by the state and you would be an heir if you are. If you are not an heir, then I am at a loss as to why you would be involved at all.

Why did you pay for the burial? This is a responsibility of your deceased cousin's estate. If you are appointed as the personal representative, then the funeral expenses are paid out of his assets. That said, yes, you can file a claim (not a lien) with his estate. Whether it will be paid in part or in full depends on whether there are any other debts/claims against the estate and whether the estate has any assets.

Like it or not, the half sister will inherit. If your cousin was really concerned and was estranged, then he would have made a will. There are services which locate missing heirs, but whether the personal representative wants to pay for this service will depend. If the half-sister really cannot be found there are procedures to handle the share of missing heirs. Her share will have to be liquidated (turned into cash) and it will be paid to the treasurer of the state or clerk to hold onto.

All of this is complex and I would recommend that you at least consult with a probate attorney in the state and county where your cousin lived prior to his death so this can be handled appropriately and you can be reimubursed to the fullest extent possible.