Depends upon what you mean by "consciousness." A lot of the literature seems to use "consciousness" just to refer to physical reality as it exists from a particular perspective, for some reason. For example, one popular definition is "what it is like to be in a particular perspective." The term "to be" refers to, well, being, which refers to, well, reality. So we are just talking about reality as it actually exists from a particular perspective, as opposed to mere description of reality from that perspective. (The description of a thing is always categorically different from the ontology of the thing.)
I find it bizarre to call this "consciousness," but words are words. You can define them however you wish. If we define "consciousness" in this sense, as many philosophers do, then it does not make logical sense to speak of your "consciousness" doing anything at all after you die, as your "consciousness" would just be defined as reality as it actually exists from your perspective. Perspectives always implicitly entail a physical object that is at the basis of that perspective, akin to the zero-point of a coordinate system, which in this case that object is you.
If you cease to exist, then your perspective ceases to even be defined. The concept of "your perspective" would no longer even be meaningful. It would be kind of like if a navigator kept telling you to go "more north" until eventually you reach the north pole, and then they tell you to go "more north" yet again. You'd be confused, because "more north" does not even make sense anymore at the north pole. The term ceases to be meaningfully applicable. If consciousness is defined as being from a particular perspective (as many philosophers in the literature define it), then by logical necessity the term ceases to be meaningful after the object that is the basis of that perspective ceases to exist. It neither exists nor ceases to exist, but no longer is even well-defined.
But, like I said, I'm not a fan of defining "consciousness" in this way, albeit it is popular to do so in the literature. My criticism of the "what it is like to be" definition is mainly that most people tend to associate "consciousness" with mammalian brains, yet the definition is so broad that there is no logical reason as to why it should not be applicable to even a single fundamental particle.
No, quantum teleportation is more akin to Star Trek teleportation whereby you disassemble the original object, transmit the information, then rebuild it using a different medium.
(More technically, you apply an operation to a qubit which is non-reversible so its original state is lost if it was not already known, but you gain enough information from this process to transmit it over a classical channel which the recipient can then use that information to apply operations to a qubit they have which places it in the same quantum state as the original qubit.)
The middle step here requires the transmission of information over a classical communication channel, and so it can't be used to send signals faster than light.
(I would go as far as to argue there is nothing nonlocal in quantum mechanics at all and the belief there is anything nonlocal is a misunderstanding. I wrote an article here on it that is more meant for laymen, and another article here that is more technical.)
There is a communication-related benefit to quantum teleportation, but not for superluminal communication. Let's say you have a qubit you want to transmit, but your quantum communication channel is very noisy. Using quantum teleportation will allow you to bypass it because you can transmit the information classically, and classical communication channels tend to be very robust to noise.
(The algorithm requires a Bell pair to be shared by the sender and receiver in order to carry it out, and so this might seem like it defeats the purpose of bypassing the quantum communication channel. However, you can establish a Bell pair over a noisy channel using a process known as quantum distillation.)