I) each NP problem can be solved in Exponential Time.
Correct. This is covered by an answer to our reference question (search for the heading "Brute-Force/Exhaustive-Search Algorithms for NP and NP$\,\subseteq\,$ExpTime").
II) if P=NP then NP=NP-Complete.
Aaaaaaaalmost true. But false. The problems $\emptyset$ and $\Sigma^*$ are the only two that would not be NP-complete if P$\,=\,$NP. These two are the problem where the answer is always "no" and the problem where the answer is always "yes". These two problems can't be NP-complete because a reduction from $X$ to $Y$ is required by definition to map "yes" instances of $X$ to "yes" instances of $Y$, and map "no" instances to "no" instances. But $\emptyset$ has no "yes" instances and $\Sigma^*$ has no "no" instances so you can't define a reduction from any other problem to one of those.
III) The following problem is in NP: given a natural number, determine whether it is the product of two prime factors.
True. This can be shown using the certificate definition of NP: a problem is in NP if there is a polynomial $p$ such that every "yes instance" $x$ of the problem has a certificate $y$ with $|y|\leq p(|x|)$ and there's a deterministic polynomial-time algorithm that, given an instance and a certificate, checks that the instance really is a "yes" instance.
In this case, the certificate is the two prime numbers. Given $x$ and prime numbers $y_1$ and $y_2$, we can check in polynomial time that $y_1$ and $y_2$ really are prime (using the AKS algorithm) and that $x$ really is $y_1y_2$.
IV) if problem X can reduce to a known NP-Hard problem, then X must be NP-HARD.
False. Informally, "$X$ reduces to $Y$" means "If I can solve $Y$, then I can solve $X$." However, if $Y$ is hard and solving $Y$ lets you solve $X$, that doesn't necessarily mean that solving $X$ was hard: going via $Y$ might have been a really dumb move. (Practical example: suppose you're hungry and have no food. If you had a Ferrari, you could sell the Ferrari and use the money to buy food. This is a reduction from the problem of getting food to the problem of getting a Ferrari. But that doesn't mean that getting food is as hard as getting a Ferrari.)