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Generally, the best time for transplanting blackberries is fall, so now would be the time to get the project done. Select a transplant site that provides full sun, air circulation and is well draining. Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/fruits/blackberries/growing-blackberry-bushes.htm

Also, it may help to do it in the evening to reduce stress/transplant shock, and be sure to include as much of the roots as possible. If you have not already done so, you may want to prune your blackberry bush as well. Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm

Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm

If they were planted this year, then it would not hurt to mulch them with straw or leaves to give them a bit of extra protection. If they are established (more than 1 year old), they do not need to be mulched.

Yes, you can usually plant them in fall as long as the soil in your area is still workable; otherwise, early spring would be preferable. Here is an article that you may find helpful: https://www.gardeningknowhow.com/edible/fruits/blackberries/growing-blackberry-bushes.htm

This article will help with pruning your blackberry plant: https://www.gardeningknowhow.com/edible/fruits/blackberries/blackberry-pruning.htm

I don't know where you're located, but I'm guessing that it's somewhere that has a winter coming, and the berries aren't supposed to flower till spring? Sounds like your poor plant's a bit confused. There's not really much you can do, except say "wow, isn't that strange?' You can just leave it - the cold will soon knock back the flowers and little fruit, and hopefully it will right itself by next spring, and get with the program. You might pull off the flowers and green fruit, to keep it from expending energy on fruit instead of storing it to get through the winter. Of course, if you don't have that cold a winter, you could just let it be to see what happens.