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In 1D dynamical systems, it is well-known that in general between any two stable fixed points there is an unstable fixed point. How does this result generalize to higher dimensions? Are there general theorems that establish a connection between the number of stable fixed points versus unstable fixed points?

Of course in higher dimension the extra complication is that we could have higher dimensional manifolds defined by $\frac{dx}{dt} = 0$, e.g. lines or surfaces. Are there general results for their stability in relation to the number of stable and unstable lines / surfaces / higher dimensional manifolds?

How about systems defined not on $\mathbb{R}^n$ but on some manifold?

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    I've personally never seen stable points topic in relation to diff geometry so disregard if this isn't useful. But what about a ball on a plateau in 2 dimension with a gravitational force field. Clearly it has stable points at the top. – 2017-02-27

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The answer is "sometimes yes". The full statement is: for some particular classes of dynamical systems there is an analogue of Morse inequalities; see, for example, this and this. For gradient systems (as far as I understand) this is just a reformulation of Morse theory for critical points.

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    Glad that you pointed this out. Indeed, as you say. Morse-Smale and gradient system have a quite related structure. Some people would say that the Conley index also qualifies... but I will forget that. The reason why I didn't mention more than "no" is that I understand the question as whether we can deduce the existence of some other equilibria from the existence of some initial ones. – 2017-02-28
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    @JohnB Oh, I've decided that the title of question is more suitable for non-negative answer, so I mostly answered to the title :) – 2017-03-01
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As you seem to have guessed correctly, the answer in no. Really the $1$-dimensional is topological: in particular you need not have stable/unstable hyperbolic fixed points, only the directions matter.