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.
There is no instantaneous information transfer (“nonlocality”) in quantum mechanics. You can prove this with the No-communication Theorem. Quantum theory is a statistical theory, so predictions are made in terms of probabilities, and the No-communication Theorem is a relativity simple proof that no physical interaction with a particle in an entangled pair can alter the probabilities of the other particle it is entangled with.
(It’s actually a bit more broad than this as it shows that no interaction with a particle in an entangled pair can alter the reduced density matrix of the other particle it is entangled with. The density matrix captures more than probabilities, but also the ability for the particle to exhibit interference effects.)
The speed of light limit is a fundamental property of special relativity, and if quantum theory violated this limit then it would be incompatible with special relativity. Yet, it is compatible with it and the two have been unified under the framework of quantum field theory.
There are two main confusions as to why people falsely think there is anything nonlocal in quantum theory, stemming from Bell’s theorem and the EPR paradox. I tried to briefly summarize these two in this article here. But to even more briefly summarize…
People falsely think Bell’s theorem proves there is “nonlocality” but it only proves there is nonlocality if you were to replace quantum theory with a hidden variable theory. It is important to stress that quantum theory is not a hidden variable theory and so there is nothing nonlocal about it and Bell’s theorem just is not applicable.
The EPR paradox is more of a philosophical argument that equates eigenstates to the ontology of the system, which such an equation leads to the appearance of nonlocal action, but this is just because the assumption is a bad one. Relational quantum mechanics, for example, uses a different assumption about the relationship between the mathematics and the ontology of the system and does not run into this.