Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
The biggest draw back (and it's a big one) is that the ring of dual numbers is not a field. It has plenty of zero divisors. So, Newton, or any of the mathematicians of the early days of calculus, certainly did not work directly in the ring of dual numbers. They of course did not consider the ring to exist (as rings did not exist at all yet), but from their writing it is clear they envisaged a field of real numbers with, somehow, some notions of infinitesimals. Their work is of course very vague, but correct. Much more on that can be found in math history books. Many interesting discussions can be found in the recent book "Adventures in Formalism", also related to the early days of calculus and how things developed.
Some (rather unsatisfactory) portions of analysis can be developed in the ring of dual numbers, but it does not go too far. The idea, as you say, is very simple, perhaps too simple. One immediately gets into trouble when trying to define the derivative as the quotient of the infinitesimal $f(x+h)-f(x)$ divided by $h$, where $h$ is infinitesimal. The difficulty is that the non-zero infinitesimals in the ring of dual numbers are not invertible. So, it's the end of the party. (As you say though, some aspects of the party remain with automatic differentiation). In some sense, the dual numbers form a first order approximation to actual infinitesimals: The square of an infinitesimal is of an order of magnitude smaller than the infinitesimal you started with, but in the ring of dual numbers, the square of an 'infinitesimal' is precisely $0$. So, in a nonstandard model of the reals you have whole layers of infinitesimals. In the dual numbers there is only one layer, nothing in it is invertible, and they all square to $0$.
The book Models for smooth infinitesimal analysis explores many different models for analysis with infinitesimals. None of them is particularly simple.
No for 1. and 3., this ring is not really useful in analysis. But it is quite important for analytical considerations in algebraic geometry, the main reason being that the scheme $\mathrm{Spec}(k[\varepsilon]/\varepsilon^2)$ classifies tangent vectors. This makes it possible to define the tangent space of arbitrary functors $F : \mathsf{CRing} \to \mathsf{Set}$ at some $x \in F(k)$, namely as the fiber of $F(k[\varepsilon]/\varepsilon^2) \to F(k)$ at $x$. There is no manifold which represents tangent vectors for manifolds, so this is the main difference.
The answers by Ittay Weiss and Martin Brandenburg are helpful. I would like to point out a more direct shorcoming of the dual numbers as far as analysis (and even calculus) is concerned is that it is not clear how to extend a generic real function to the dual numbers, even say a $C^\infty$ smooth function. Thus, if one wishes to form a ratio of infinitesimals involved in the definition of the derivative, it is not clear what should appear in the numerator. Over the hyperreals, one has a systematic way of extending every real function to the wider hyperreal domain, and the transfer principle (which is arguably a formalisation of the Leibnizian Law of Continuity) ensures that such an extension is meaningful.
For this reason, the answer to the original question would be: No, dual numbers are insufficient to capture "Newton's historical method for doing calculus". The hyperreals provide a framework where the procedures of 17th century infinitesimal calculus can be successfully formalized.