Regards to L5 being the main feed forward pathway going up the cortical hierarchy: This idea originated with Murray Sherman at the U Chicago. He has been promoting it for many years. He has written multiple papers on this topic and it is featured heavily in his most recent book. I know I read at least one other paper from another lab that specifically tested this hypothesis with a positive conclusion. Unfortunately I can't recall the authors. I suspect with a bit of literature searching you can find some relevant papers. You can always email Murray and ask him the state of support.
Everyone agrees that both L2/3 and L5a project to the next region. The question is what are their relative roles. Sherman argues that the L5a pathway, which goes through relay cells in the thalamus, is the main "driver" pathway. Although the number of axons is numerically smaller than L2/3, those axons are large and form large synapses on the recipient layers L4 and L6a. Hence they can make the recipient neurons fire. The axons from L2/3 are numerically greater but form smaller synapses that are not proximal, therefore they play a "modulatory" role. Another supporting piece of evidence is that primary sensory regions only get input via the thalamus; there is no equivalent of a feed forward L2/3 path. If the primary driver input to primary sensory regions is via the thalamus then it would be very odd if that wasn't true of higher regions which exhibit the exact same pathway and anatomy. Another piece of evidence is that the L5a/thalamic neurons seem to only target the higher region, whereas L2/3 cells project many places, not just to the next higher region.
We did some literature searches on the question of long range connections into L6 between "what" and "where" regions. Thomson stated these exist in her review paper. I read the papers she referenced and a bunch of others. The anatomical evidence was not conclusive. There seems to be a general level of agreement that what and where (motor and sensory) regions are highly interconnected. There seems to be evidence that most of these connections arrive in the lower layers. But I couldn't find out which L6 cells receive this input. I also found evidence that these connections originated in multiple layers on the "where"/motor side, which somewhat contradicts what Thomson said. As is often the case, the different studies used different animals and protocols. This makes it extremely difficult to reach definitive answers.
I am sorry that I can't cite specific papers, but as you know I have no memory for authors. It isn't hard to find these papers with a bit of searching on Google Scholar or elsewhere.