I'm in the process of cutting my black raspberries now - they're the ones pictured. My preference is to "tip layer" them, essentially leaving long canes with the tips pinned to the ground in early spring for cutting free around now. I leave other canes for fruit with a "heading cut" to force the side branches where the fruit forms.

I like fall planting better, but I'll qualify that with "for temperate climate areas". For New England in particular autumn is when insect pressure decreases, precipitation is a bit more regular, and temperatures are kinder to plants. Plus there's additional time in the ground for the plant to establish, and it's my belief that the warmer soil temperatures relative to air temperatures helps to drive root growth even while the aboveground sections are dormant.

I definitely will!

For cane fruits like raspberries and blackberries, no. Truth be told I avoid rooting hormone liquids and powders when I can, they can have some nasty health effects without PPE. Many different fruit bushes can be cloned either by rooting the tips, mounding around the base (stool layering), or by dividing rooted stems without the need for chemicals, but others can be rooted in water with willow cuttings - the willows will leach a safer concentration of the rooting hormones into the shared water.

I leave little pockets of extra mulch around the tips I'm planning to take and replant as a visual cue, but stones or logs or some other kind of weight will help to keep them in place for rooting. Older canes from bushes you like can make good cutting material, but can be more difficult to pin without breaking. Some varieties of raspberry will only fruit on first year wood, and for those I do think it's totally worth pinning right after fruit harvest for additional first year plants.