There is no problem planting any of the fruits you mention in proximity to each other. Cross-pollination only comes into play if you allow seeds from the fruits to sprout and then let the seedling plants bear fruit. Fruit from the original plants will be unaffected.

Crosses of raspberries and blackberries and of currants and gooseberries are done all the time intentionally by plant breeders.

As mentioned there should be no issue with the varieties you have mentioned. Though when talking about blueberries there is potential reason to not plant rabbit eye and high bush varieties together, at least in commercial production situations.

How wide is the strip? If you put 6 blackberries into a 35 foot narrow bed I don't think there would be room for much else. A blackberry takes up a lot more than 3 feet. More like 6 at least. Have you got wires in place for training?

Most planting guides for thornless blackberries suggest a spacing of about 3 feet between plants, though using wider spacing does not change things much since blackberries will send up root shoots, or create daughter plants where canes touch the ground, filling in the gaps over time (likely within 2-3 years).