To answer the title question, yes, computation is independent of hardware. Computation is defined by the transformation of information, not by how it's embedded in the real world. This is easy to see: all the computation models have a mathematical formulation, and you can write mathematics as symbols on paper. In fact, many models of computation, such as the lambda calculus (1936) and Turing machines (1936) were invented before there were general-purpose computing machines (there were numeric calculators and data sorters, but programmable machines didn't exist until the 1940s).
It is possible to model today's computers by mathematical tools. They're just finite-state machines, after all. There are a lot of states, so a complete pen-and-paper representation is not feasible, but if you only model the part that you're interested in, that can be within reach.
As to whether there is only a single notion of computation, we don't know. We think so: that's the Church-Turing thesis. We think so because we've invented a lot of models of computation, and they all turned out to be equivalent. Some of them are easier to work with than others, or more directly applicable to certain problems, or are a closer match for certain computing hardware (e.g. in terms of performance or locality). But they can express the same computations.
Are Turing machines (or any of the equivalent concepts) the last word on the topic? We don't know. We certainly have weaker models of computation, but we think of them as restricted computation. And we have stronger models of computation, but we think of them as “magic“ — we can reason about them, but we have no idea how they could be implemented.
Is the brain a Turing machine? That's a question about the brain, and we don't know. We know that it's at least as powerful as a Turing machine, since it can simulate one. We are very far from being able to accurately simulate a brain with a Turing machine, but we don't know whether that's a matter of scale (a brain has tens of millions of neurons — a PC doesn't have enough memory to even store all the connexions). We do tend to think of brains as being more powerful than computers — when a problem is undecidable (in the Turing computation sense), we offer to solve it by thinking, but that doesn't answer the question. When we solve a problem by thinking, the solution is expressible in Turing machine terms. The problem solving method, however, may or may not be more powerful than Turing machines.
Would aliens have the same notion of computations? We don't know. We think so — mathematics seems universal to us — but that's a question about aliens, and since we don't know anything about aliens except what your imagination tells us, we can't answer it.