Computation should be treated more as something formally defined by a Turing machine, not that the Turing machine is defined by computation. For thousands of years computation was this weird arbitrary thing that humans did on pen and paper that was able to get results, Turing merely formalized that into a very simple analog of what it means to compute something.
Computability of course is defined using the Turing machine as such. The fact that a Turing machine can do something means its computable, perhaps there is a more abstract idea of what it means for something to be computable but for now this is what we have.
There are things a Turing machine cannot compute of course. Turing machines for instance inherently cannot produce randomness, they are deterministic machines and it was decided that determinism was to be a property of computation and as such the Turing machine is restricted to being such.
Able to do conditional checks
You say this as if its a simple thing but there is a lot more basics you need to even get to this step. A condition requires a notion of true and false, which isn't a given. A condition at the very least requires a notion of equivalence as well, which isn't guaranteed either and needs to be defined.
It also needs a basic set of things it can do upon the true or false of some condition. You forgot to include that in your properties.
I don't know if you have ever programmed yourself before but conditionals, jumps, are really all you need. Loops are merely if and jump statements wrapped together, functions are similarly jump statements that are specially arranged. All of this translates down to what you say.
Our best knowledge of physics right now still showed that everything can be computed with conventional algorithm. Any systems that follow current physics can be - in theory - simulated by conventional algorithm, so completely within the reach of a Turing machine. This includes the human brain.
For a physical system to perform computation beyond the reach of Turing machine, it has to follow new physics that we have not discovered.
Physics had already gave us interesting new possibilities beyond the conventional algorithm: quantum algorithm. Unfortunately, quantum algorithm is merely (maybe) more efficient than conventional algorithm. So from a computability perspective ("can this be computed at all, no matter how long it takes?"), quantum algorithm is not better than conventional one.