The book Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Vol. 343 by Bahouri, Hajer, Chemin, Jean-Yves, Danchin, Raphaël contains plenty of information on basic Fourier analysis.

The Fourier multiplier $a(D)$ is the convolution with $\hat a(-x)$. For that operation to send compactly supported functions into compactly supported distribution, it is sufficient that $\hat a$ is compactly supported since $supp (u\ast v)\subset supp u+supp v$. It should also be necessary since $\hat a(\zeta) \hat u(\zeta)$ must be entire of exponential type when $\hat u$ is entire of exponential type.

Geometrically speaking your parabolic equation is $X+Q$ where $X$ is a real vector field and $Q$ a positive elliptic operator of order 2 (e.g. $-\Delta$). You may note the following equality of vector fields: $$\partial_t+\partial_s=\partial_\tau.$$

Drastic differences between Schrödinger and heat equations: the heat equation is an hypoelliptic diffusion equation whereas Schrödinger is a propagation equation whose speed depend on the magnitude of the frequency and of course Schrödinger equation is not hypoelliptic.

The function $x\mapsto\det(u_1,\dots,u_{N-1},x)$ is continuous and $\mathbb H$ could be taken as $span(f_1,\dots,f_{N-1})$.