Does the Ryu-Takayanagi conjecture only apply to vacua? Does the Ryu-Takayanagi conjecture only apply to vacua, or does it also apply to arbitrary excited states?
For excited CFT states, the entanglement entropy can be proportional to the volume, not surface area, or it can even be arbitrarily large.
 A: The Ryu-Takayanagi (RT) formula certainly applies to more general backgrounds, but it doesn't hold for arbitrary excited states. Remember that generic high energy excited states in the CFT will correspond to a bulk which isn’t at all semiclassical and will look very quantum, i.e. no nice bulk geometry. But of the CFT states that correspond to semiclassical bulk geometries (i.e. quantum fields with classical gravity), the RT formula will apply.
Remember that RT says that the entanglement entropy of a boundary region is equal to the area of a minimal surface in the bulk. But if the bulk is an AdS black hole, that minimal surface can pick up an extensive contribution from the black hole horizon. The larger the black hole in the bulk, the larger this extensive contribution will be. You are correct that entanglement entropy can be proportional to the volume (e.g. take a random state in your Hilbert space), but these will be the states that are very quantum in the bulk.
Regarding RT in specific bulk geometries: The original paper by Ryu and Takayanagi discusses some more general backgrounds other than the AdS vacuum. Also see this review. More general (time-dependent) backgrounds are discussed in the covariant generalization of RT. Also, different types of corrections to the RT formula are known (both quantum corrections and corrections from higher derivative terms), which I think are explained in Sec 5 of this review.
Also, the RT formula isn’t really a conjecture (at least no further than holography is a conjecture) as it has been derived from the bulk gravitational path integral in this paper.
