Actually, the 4~5 layer newspaper/1 cardboard layer helps to keep any weed seeds as well as root-spreading weeds in the mounded earth from growing.
By the end of the season, the paper layer would have broken down, but by that time, all the mulch will have helped to suppress the weeds and you won't need to renew the paper layers. Just keep piling mulch.
The Winter cover crop issue (as well as self-reseeding crops) is something I've struggled with. I have to admit I don't mulch as much and all over for this very reason. You'll note in the video and text (you'll find links in that Hay thread) that Emilia Hazelip starts a lot of her plants in the green house using compost. Like her (and Ruth Stout), I've found that big seeds like beans, cukes, squash, etc. will grow if mulch is pushed back and planted in soil, but where the mulch is too thick and less broken down, pushed back mulch mysteriously covers over the area again* and prevents -- especially smaller peas and beans from -- sprouting. Creating pockets in the mulch, filling with potting soil or soil mix and planting seeds works.
* Aside from the mulch naturally straightening out from being pushed aside/bent, I suspect Robins and Grackles as well as Moles and Chipmunks toss the mulch about. Chipmunks and Grackles will also EAT the seeds if they find them.
My speculation on how to go about growing cover crops:
On a newly prepared mound, to sow densely broadcasted covercrop, simply rake smooth with the back of the rake to even out the competition (I.e. nothing else has a head start), then broadcast a FAST GERMINATING covercrop like Buckwheat or Clover.
For winter covercrop after a season of use, I think the best bet would be to NOT add any more mulch/remove some mulch so there is less than 2" layer prior to sowing the covercrop seeds. BEFORE the season is over (at least 4~6 weeks to give the covercrop chance to germinate and grow), broadcast large-seeded/strongly germinating covercrop like Winter Field Peas or grain like Winter Rye or Oats among the existing and still producing plants. When the production plants are done, cut rather than uproot the plants.
... here I have to admit I see a stumbling block. According to M. Fukuoka, it's OK to step on two week old germinating barley, but what about older covercrop plants? Also, would the winter-type covercrops continue to grow past frost until freeze?