Why is the halting problem unsolvable by a turing machine?
It can be proved that the halting problem cannot be solved by any Turing machine. There are several question regarding this on this side which you can look up.

It is not true that humans can determine if a given Turing machine halts. For example, we have been unable so far to compute many values of busy beaver functions, which measure the maximal number of steps a halting Turing machine can run given the number of tape symbols and number of states.

According to the Church–Turing thesis, every machine can be simulated by a Turing machine. So DNA computers, quantum computers and the like will not help. All they can do (especially the latter) is to dramatically speed up computation, but they cannot help computing non-computable functions.

There is a field known as hyper-computation which studies machines stronger than Turing machines, but these are not believed to be realizable by most practitioners. Recursion theory explores such models from a more theoretical perspective.