Can I dry the fruit out to make a plant?
Not really. The fastest way to make new blackberry bushes is by taking root cuttings. This should be done in January or February, when the plant is dormant. This article will help with that: https://www.gardeningknowhow.com/edible/fruits/blackberries/propagating-blackberries-cuttings.htm
Is it safe to put cedar mulch in a bed of blackberry and red current bushes? If not, what can I use to keep weeds down?
It is safe to use it, but some people feel that the fact that cedar decomposes so slowly, that it can harm the fertility of the soil. A bark mulch or hardwood mulch will also work.
Is it necessary to, and how does one, prune blackberry bushes? Also, we live in a cold area and some vines die off during winter, is this normal?
Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm
Winter dieback is normal, and plants should regrow in spring.
What kind of fertilizer should be used and how often?
Use an organic fertilizer, such as compost, which can be applied during planting. You can also mix some with water and apply during watering about once a month, although this isn't a requirement.
We have a blackberry bush that has orange pollen-like spots on the leaves. They are only on leaves that do not have any flowers. Is this a pest? What should I do about it?
It is most likely plant rust. This article will help:
https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm
Planted two bushes last year and have lots of berries ripening. There are three very tall stalks growing. Do I leave them alone?
If they are growing well, you should let them be as they are.
I have inherited some blackberry bushes on my allotment. Both varieties are thornless and I pruned according to instructions. However, I now have some very vagrant new shoots coming up, which have thorns. These shoots are very strong. Why is this, and do I need to cut them back to ground level?
Some varieties of blackberries are blended with others to create a blackberry that is thornless as the plant bought, but that will revert to a thorny blackberry if propagated. These sucker are a form of propagation (though not intentional), so they grow with thorns. It would be best to cut out the shoots that have the thorns.