The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts.

1. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifolds, where almost closed means that the boundary is a homotopy sphere.
2. Understand those homotopy spheres that arise as boundaries of almost closed compact oriented smooth $(n-1)$-connected $2n$-manifolds.
3. Understand inertia groups of smooth oriented closed $(n-1)$-connected $2n$-manifolds.

In the work you mentioned, Wall achieved a classification of type 1. in terms of what he calls n-forms. Since then several authors (Wall, Kosinski, Schultz, Stolz, ...) have obtain partial results regarding 2. and 3. Most recently Burklund--Hahn--Senger and Burklund--Senger settled the last open cases and completed the classification.

The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts.

1. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifolds, where almost closed means that the boundary is a homotopy sphere.
2. Understand those homotopy spheres that arise as boundaries of almost closed compact oriented smooth $(n-1)$-connected $2n$-manifolds.
3. Understand inertia groups of smooth oriented closed $(n-1)$-connected $2n$-manifolds.

In the work you mentioned, Wall achieved a classification of type 1. in terms of what he calls n-forms. Since then several authors (Wall, Kosinski, Schultz, Stolz, ...) have obtain partial results regarding 2. and 3. Most recently Burklund--Hahn--Senger and Burklund--Senger settled the last open cases and completed the classification.

Regarding your second question: In the topological category, 2. and 3. are vacuous since all high-dimensional homotopy spheres are topologically trivial. Wall's original approach to 1. only uses tools that have since been established in the topological category (mostly by Kirby--Siebenmann), so his approach goes through and reduces the classification to understanding $\pi_n(BSTop(n))$. To get at the latter, you can compare $BSTop(n)$ to $BSTop$ and use that its homotopy fibre receives a highly-connected map from $SO/SO(n)$.