Is topology just a generalization of real analysis?

The structure of $\mathbb R$ is richer than its (usual) topology. Its topological properties would make it "metrisable" but would not provide it with a metric, and you need a metric to do worthwhile analysis. The (usual) metric on $\mathbb R$ is usefully compatible with its algebraic properties.

There are topologies on $\mathbb R$ other than the usual one, and alternative metrics also exist. So analysis in the usual sense cannot be reduced in a simple way to topology. The particular useful structures on $\mathbb R$ are particular, and useful.

One (helpful) way of looking at topology is as trying to capture the essence of continuity (which can be expressed in terms of open sets). This is obviously important in analysis, but isn't the whole story.

Yes. In fact, the morphisms of a topological space are continuous functions.

To proceed from real analysis to topology we must do a few things.

1. Abstract the distance formula in $\mathbb R^n$ to a general metric. From this we go from real euclidean spaces to metric spaces.
2. Using the metric space definition of continuity, we prove that a function is continuous if and only the inverse of an open function is open. This theorem eliminates the need for the metric in the definition of continuity.
3. We then use the closure properties of concrete open sets in metric spaces to define a topology and from that we get a topological space.