In corn there are a couple of style of headers - most of the newer ones have what are called stripper rolls under the header which are rollers that grab the stalk and pull it vertically and in doing so essentially shred the stalk. The other style was where the forward motion of machine forced the stalk into the header and the ears couldn't fall through and were caught by rollers or chains and pushed into the gathering auger. In this case the stalks are left more bent over and not shredded. Dryer climates with snow cover usually prefer the second style for less wind erosion, plant matter loss due to winds, and catching more snow.
The planters for either of these situations are the same where they have spiked wheels called trash wheels in front of the vertical opening disc that remove the lion's share of the remainder so that disc cuts through and the new seed is planted at the right depth in soil. You can hardly tell if the field is planted until the new crop starts coming up but even then the field looks like a mess. But by the time it comes around to harvesting again most of the stalks have rotted to the point that they are not standing and have fallen over on the ground and probably 80% are humus. So the stalks are not a real problem unless you want to till the ground as the stalks - especially with the second scenario above - will bridge up in tillage equipment and make a mess which is why a lot of people used stalk shredders until the headers started taking care of it. (BTW - the headers with the stripper rolls push through the field much easier as there is not the drag on the head especially with the larger headers.)
The second part of your question is the weed control. Normally in no-till farmers plant and then spray prior to the crop emergence with a Round-up type herbicide to knock down anything that is out there. Then there is a lot less weed growth for two reasons - 1) without tillage only last years weed seeds matter all those that are underground stay whereas tillage turns them up and some seeds will stay underground twenty years and get turned on top and grow again, and 2) there is a lot of mulch if you will that slows weed growth until the crop gets up to shade the ground and stop weed growth by not allowing the sun to get through. Still, while getting the system going for the first few years a second application of herbicide is typically applied to the growing crop. Since my nephews have been doing this for 25 years though unless there is a strange weather pattern that slows the early crop growth they do not apply a second herbicide.
So what we have done here is taken what is normally and early spring tillage pass to remove trash, a second tillage pass for seedbed prep, a planting pass, a herbicide application to stunt the weeds and let the crop get ahead, a harvesting pass, and a fall tillage pass, i.e. six trips over the field 3 with high horsepower and fuel usage, and reduced it to a planting pass, a herbicide pass, and a harvesting pass, i.e. 3 passes and none with high horsepower. Also we left variables that were common but as much without the tillage of passes for insecticides, fungicides, fertilizer application (often with planting but with tillage often a second application like anhydrous).
Note: One of the big concerns of many farmers in the corn belt is all the trash on the soil not letting the moisture out and the sun in for a combination of warming and drying the soil to be able to get on it in the spring thereby delaying planting and slower early crop growth. My nephews in SD have found they are delayed a little but not significantly so even with their fairly short growing season they are minimally affected by this but in a less dry climate it is a bigger concern which may pave the way for smaller autonomous machinery that can float across the wetter soil and can work 24 hours a day and still get everything accomplished. The future will be interesting.
I've got family in Onida, Highmore and Chamberlain.
