Pyramid vs Scale-space in Focus Stacking
Your question relates to techniques with different names. The  first one is related to micrographs taken at different (aligned) focal positions combined  into a focus image. This Extended Depth of Field is based on complex wavelet-based methods.

To answer your second question on difference between MRA, pyramids, etc.: most pyramids and wavelet-based methods for 2D image processing are instance of multiscale/multirate//multiorientations  decompositions based on certain scale-space discretizations (a review paper on these issues is A Panorama on Multiscale Geometric Representations, Signal Processing, 2011). With scale-space in general, you do not need to downsample, and end up with a stack, not a pyramid.

Since one selects maximum coefficients across scale, band, or orientation, my opinion is that a complex pyramid, a complex (or dual-tree) wavelet transformation, with a little redundancy and some shift or rotation invariance could be useful, at the cost of more painful implementation. Standard real MRA wavelets are limited in orientations, full scale-space might be quite involved.

Maximum coefficient selection is probably  a poor's man saliency detection, which is another track to follow based on the specificity of your images.

More general terms for your search are multifocus image fusion, multiscale or multi-resolution images fusion, even multichannel. A few references:


Performance comparison of different multi-resolution transforms for image fusion, Information Fusion, 2011
Multifocus image fusion and restoration with sparse representation, IEEE Transactions onInstrumentation and Measurement, 2010
Multifocus image fusion using the log-Gabor transform and a Multisize Windows technique, Information Fusion, 2009


You can as well find inspiration in multiscale versions of High-dynamic-range (HDR) imaging, for instance:


Local Edge-Preserving Multiscale Decomposition for High Dynamic Range Image Tone Mapping, IEEE Transactions on Image Processing, 2012
Multiscale Processing with HDRWaveletTransform