The: If $\text{P} \neq \text{NP}$ implies NP-Hard problems require exponential time algorithms is a common myth, which seems quite prevalent.
In fact, we have known NP-Hard problems which have sub-exponential algorithms! For instance see this here: http://hal.inria.fr/docs/00/47/66/86/PDF/cops_journal_2008_10_16.pdf
That said, there is a conjecture, called the Exponential Time Hypothesis which states that 3-SAT (and some of it's variants) cannot be solved in sub-exponential time.
Now you might wonder why does the above sub-exponential algorithm for an NP-Hard problem not falsify the Exponential Time Hypothesis, as we can ultimate form a reduction from 3-SAT?
The reason is that reductions are allowed to blow up the input polynomially. Say $A$ reduces to $B$ and $B$ has a $\theta(2^{\sqrt{n}})$ time algorithm. Now if the reduction makes the size of the input to $B$ as $n^2$ (it was $n$ for $A$), that still gives us a $\theta(2^n)$ time algorithm for $A$. So, even though $A$ never has sub-exponential time algorithms, we could still have it reduce to a problem which has sub-exponential time algorithms.
As to why we have non-determinism, I am guessing this was first started by Rabin and Scott in their papers about deterministic finite automaton and non-deterministic finite automaton.
Once Cook/Levin introduced the concept of NP-Completeness and Karp et al gave a seminal list of 21 natural problems which are NP-Complete, NP became a very important class.
I believe it was Edmonds who first characterized NP as 'problems with polynomial certificates'.
(Note: The above three paragraphs are from memory and I haven't verified them).
In fact, there are characterizations of NP which do not use non-determinism. The current hot topic has been the use of PCP Theorem to prove various NP-Hardness results for approximation versions of NP-Hard problems.
So, non-determinism is still part of the definition of NP and is historically important, but the current status is that it is not mandatory.