Just jumping in here quick. Having problems with the board and still getting positives on anti-virus and malwarebytes.....
Appeals are automatic. Unless a plea deal is in place that she agrees to, in order to drop appellate rights.
Yes, the State will have to pay for them. They will not be successful, they will just be paper work that requires a couple of years.
She already admitted the murder. Any issues with the Judge's ruling are going to be very much a waste of time, but required in every death penalty case. I think Judge SS has handled everything quite well, and to tell all the truth, not many appeals ever make it. Unless an attorney or Judge openly says out loud that they compare the defendant to Charles Manson or Ted Bundy, or the DNA comes back later that more were involved. (They have hers at the crime scene and her admission of murder.) Pointless as it may be, expect it. And, don't bother to hold your breath for a couple of years. (At this point, nothing merits a reversal of the conviction.) ETA: Also know that whomever is assigned to do the appeal must bring up some legal issue, no matter how weak. Do not fret, it is their job, and the know it is fruitless, but they get paid for it. It all looks good on paper, but the lawyers know, the Judges know, and eventually the Justices know, that it is just part of the job. It will not be granted. It will be some minor something, but that is what is required. Never forget, she already admitted the murder and know, it is fruitless. Never forget the evidence.
Also, why would anyone watch the made for t.v. movie? It is offensive! Oh, I don't have cable. Guess it is normal, but the less you tune in the more these crappy movies stop. Digital world will know if you do watch, not kidding. Digital t.v.s, t.v. boxes, not being uber paranoid, just being honest.