hau Charlie, welcome to Permies.
As far as straw goes, what kind of straw? Wheat, Barley, Rye straws generally do not have herbicide sprayed on them prior to harvest, it simply isn't needed to defoliate since the plants die as the grains mature.
Rice straw on the other hand, always tends to be sprayed with defoliant since the plants don't die as the grains mature (you can actually double crop rice just by harvesting with out spraying any herbicide).
If you are worried about cides, soak the bales of straw completely before you use them, this will leach most of the residuals out.
A mulch layer around 3 inches thick will do the job of weed suppression yet still let your desired seeds germinate.
Daikon radish is one of the best large
root vegetables to use for de-compaction of soil the
roots can get huge which will put back lots of humus as they decompose in the soil. They also attract good fungi and microbes.
If you are trying to improve the soil quickly, broadleaf plants, while good, will shade thin leaved plants which could stunt their growth. We plant in stages so all plants have equal opportunity to sprout and grow.
You also have to account for length of time from planting to germination, the plants with longer germination times need to go in first and you work your way to the seeds with the shortest germination time.
Multi plant type cover cropping means you will need to do some planning to get it right. No-Till is actually fairly easy and if you have a crimper, cutting isn't required.