Examples of processes / problems that cannot be tackled by Turing Machines
I'll address as answer only part of the question (see my comments for why not all of it). The interesting phylosophical part of the question is basically asking if the Church-Turing thesis (CTT) describes all that happens in the universe. This is much more a physics question than it really is a CS question. CTT has been extended to the Church–Turing–Deutsch principle to account for quatum computers; it basically states that a quantum computer can simulate any physical process in the universe.

So what about continuous processes that we normally model over $\mathbb{R}$? Well, there are two aspects of this: 


Models of computation with infinite-precision real-numbers have been devised. Whether you think such a model is the "right" one for physical processes, depends on physics and...
There are tough issues like the Bekenstein bound, which limit the amount of information that can exist in a finite region of space as we currently understand it. For details on this latter issue, it's much better to ask on our physics sister site.


A bit more info with pointer to additional readings is found at https://mathoverflow.net/questions/54820/physics-and-church-turing-thesis