The coordinate-free concept behind transposes is the adjoint, which shows up in multiple places in mathematics. Basically, you have a formula:

<Ax, y> = <x, By>

where <,> is some inner product, and you say that the the linear transformation A is adjoint to B. If you take the usual inner product, then A is B^T.

A previous conversation: https://news.ycombinator.com/item?id=15948153

As for 2, norms (and general inner products) show up in more theoretical parts of math that use linear algebra. Maybe an applications-focused treatment doesn't need to focus so much on it?