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The raspberry and blackberry plants are susceptible to the same diseases. Planting them separately will help. Also, keep your berries planted away from your vegetable garden, if possible, as these are also susceptible to some of the diseases that affect tomatoes and other nightshade plants. It would probably be better to put your blackberries and raspberries elsewhere anyway since they need quite a bit of space to grow. These two berry plants should also be planted some distance apart. Wild blackberries are most likely to carry the fungus that affects these plants, so look for berry plants that are domesticated varieties and certified virus free.

There are several reasons that a blackberry might not grow fruit that you can read about here: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberries-not-fruiting.htm

Yes, it should be ok to cut out any dead growth. This article will help with pruning these bushes: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm

First-year canes, also known as primocanes, appear as stems that arch or trail along the ground. They usually bear large compound leaves with five or seven leaflets; they do not produce any flowers or fruit. Second-year canes grow to become what is known as a floricane. The stem will no longer grow taller, but will begin to produce lateral buds and flowers. Also, these normally have smaller leaves with only three or five leaflets. For information on pruning, this article should help: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm

The most important things you can do are making sure the bush has plenty of water, and cut the branches back 1/3 - 1/2. This article has more information: https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Most types of weeds have a type specific weed killer that kills only that kind of weed. I would suspect that your weeds are either grass types or broadleaf (e.g. dandelion) type weeds. You can purchase the weed specific type of weedkiller and this will kill the weeds without hurting the bushes.