Subsampling property of wavelet transform
I am not completely confident with the full theory of wavelet frames, however I will share some pointers and my beliefs, do not take any of  the following for a truth. The property you show is (partly) a characterization of an orthogonal system. In Hernandez &amp; Weiss, A first course on wavelets, 1996, you find Theorem 2.12:

If $\psi\in L^2(\mathbb{R})$ is a band-limited function such that $\hat{\psi}$ is zero in a neighborhood of the origin and $\{ \psi_{j,k}: j,k\in \mathbb{Z}\}$ is an orthonormal system, then this system is coomplete if and only if 
$$\sum_{j\in \mathbb{Z}} |\hat{\psi}(2^j \xi)|^2=1 \quad \text{a.e. on }\mathbb{R}-\{0\}$$
and 
$$\sum_{j=0}^{\infty} |\hat{\psi}(2^j \xi)\hat{\psi}(2^j (\xi+2k\pi))| =0\quad \text{a.e. on }\mathbb{R},\quad k\in 2\mathbb{Z}+1$$
with the system defined as:
$$ \psi_{j,k}(x)= 2^{\frac{j}{2}}\psi(2^j x-k)\,.$$
So the conditions are a little bit stricter for an orthonormal system. 
So it seems rather to me a characterization in 1D (f band-limited wavelets), than can help define radial 2D wavelets, than a characterization of radial wavelet per se. So if I understand your first question well, this imply the possibility of discrete wavelets and downsampling.

Since it is a characterization, I (very unsure) would say that you can downsample, but you have no guarantee about the completeness. I do not understand your condition  $$h(\omega)=1, \forall \omega&gt;\pi/2, h(\omega)=1, \forall\omega&gt;\pi/2,$$
but I can add that in Chapter 3.4 of the above book, you have characterization of wavelets with support in $[-\frac{8}{3}\pi,-\frac{2}{3}\pi] \cup [\frac{2}{3}\pi,\frac{8}{3}\pi] $, so I would say yes, as $\frac{8}{3}&gt; \frac{1}{2}$. I would say that the limit on $\pi/2$ is not an issue because of the dilations of the mother wavelet.

For the last question, without further information on the $h_i$, it is unlikely that you generate a wavelet system (possibly because of refinement equations), but as long as you cover the whole spectrum, who really cares? You get a decomposition that is energy preserving, but I am not sure yet it suffices to define a tight frames.

Additional lectures:


Affine Density, Frame Bounds, and the Admissibility Condition for Wavelet Frames, 2007, Gitta Kutyniok, with results in frames with generalized samplings  $ \psi_{j,k}(x)= a^{\frac{j}{2}}\psi(a^j x-b)$
A first course on wavelets, 1996, Eugenio Hernandez and Guido L. Weiss
Ten Lectures on Wavelets, 1992, I. Daubechies, esp. Chapter 3
Tight Wavelet Frame Construction and its application for Image Processing, 2005, Kyunglim Nam
Frames and bases. An introductory course, 2008, Christensen, O.