Sunflowers naturally bend downwards as the flowerheads start to mature seeds. It protects the seeds from rain and dew standing on the surface, which would cause mold.
The seeds are harvest-able when they are hard to a fingernail, and have attained their mature color. Note that each "sunflower" is actually a flowerhead, with many individual flowers. Those in the center are small and inconspicuous, but you can see the pollen on them as they bloom. The yellow petals are on some of the flowers at the edge. In the center, you can see the flowers that are pollinating are in a ring, maturing from the edges inward. First they open and develop pollen, then dry and shrivel, then the swelling seed becomes visible, and eventually the seeds develop their color, size, and hardness of maturity. One sign of maturity is that the calyx (the back of the flowerhead, that holds all the individual flowers) turns yellow and then brown. When yellow, some seeds are likely mature; when brown they all are (or as many as there will be, since sap is no longer going to them.)
Another sign that they are approaching maturity is that birds show up to eat them. (Birds learn after the first year, and show up in great numbers. This is a good reason to skip a year between sunflower crops, or move them to a different part of your
land.) In order to get your seeds to full maturity, you may need to use some kind of barrier--individual bags or a temporary hoophouse covered with fabric, netting, or plastic. Note that if you use netting with holes large enough to work a seed through, the birds will try to do so. So any netting with holes needs to be held above the flowers too far for birds to reach in and peck.. We covered our entire bed last year with agribon (reemay) on t-posts and hoops. If you have a small number of plants, bagging is best. Paper bags will work in rain-free climates, but it's better to have something that allows more airflow. fine nylo//n mesh is ideal. You may be able to get it cheaper at a fabric or crafts store than at a garden/hardware supply. Note that agribon or reemay can be sown just like cloth with a hand needle or a sewing machine and make very economical reusable bags.
Since the flowers within the flowerhead don't bloom at once, they won't mature all at the same time either.
When to harvest is a judgement based on the number of seeds that are fully mature versus the chances of losing them all to birds, frost, or heavy rain.
We cut them, then pile them in baskets,
milk crates, or similar containers that allow airflow, and suspend those from the roof of the greenhouse to dry. An onion sack made of netting would work, but you might lose seeds through it. I like open baskets with a handle best.
Do NOT pile them on a table or on the ground. Mice and birds love them. I have even known mice to run out along roof beams and climb down a rope to a rack attached to the ceiling. Here is our method: We run a clothesline or wire across the room or greenhouse. Then the container of flowerheads is hung from that. So far, we have not encountered tightrope-walking mice.
Once they are completely dry, you can get the seeds out by rubbing on a coarse screen or a piece of 1/2" to 3/4"hardware cloth mounted on a frame. I use the same one we use for screening
compost and soil. Or a more laborious method is just to rub and bend the heads with gloved hands. Do wear leather gloves for either of these operations. The rubbing will loosen both the seeds and the stiff bracts that hold the flowers in place and form the honeycomb structure of the head. You'll need to separate out the seeds by winnowing. They are heavier than the bracts, so pouring from on high into a container in front of a fan will do the trick. Alternatively, there is a way of putting them into a big salad bowl, and jiggling so that the heavy seeds sink and the bracts collect on the surface, like gold panning. The bracts and other chaff can be skimmed or blown off the top.
Do either of these operations on a tarp--they take some learning, and you don't want to lose your seeds irrevocably. For hulling, I don't have
experience, but the Davebuilt hand-crank nutcracker is adjustable to many many sizes and might well work. Other possibilities includ a corona grain mill with the plates very loose and an old-fashioned laundry wringer.
One variety well worth growing for its beauty, longer bloom, use as dye, and high oil content is Hopi Black Dye. This is also a great choice around
chicken runs, as they adore the seeds. If you throw them the occasional seedhead, it will entertain them for a long time. The heads can also be dried and fitted with a wire or ribbon to hang as a bird feeder in winter--these make great gifts. for oil, you can use the Piteba hand-crank press, or do as the
Natives did and crush them, then boil the whole mass for a long time, then skim the oil off the top. The seeds are also edible as snacks, just not as large as the confectionery types.
https://www.quailseeds.com/store/p264/Hopi_Black_Dye_Sunflower.html
Which brings us to sunflower seeds for eating. THE kind to grow is Humongous if you want big, easily-shelled seeds. The seeds average 3 times greater in mass than Mammoth, and are of a similar shape--rounded, not long and skinny like the Mongolian Giant. It is also the easiest to grow of the giant sunflowers (Titan, Mongolian Giant, etc.) A farm near us trialed every giant type available, and Humongous had the biggest seeds, best germination, fastest growth, earliest maturity, and best yield.
https://www.quailseeds.com/store/p72/Humongous_Sunflower.html
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I’m trying something new this year with the survivors; I’m covering the seed head with shade cloth towards the end of the flower bloom time. Keeping my fingers crossed. 
